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CONFIDENTIAL 





Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Groups of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol¬ 
ume was printed and bound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has been 
made by the War and Navy Departments. Inquiries concerning the 
availability and distribution of the Summary Technical Report 
volumes and microfilmed and other reference material should be 
addressed to the War Department Library, Room 1A-522, The 
Pentagon, Washington 25, D. C., or to the Office of Naval Re¬ 
search, Navy Department, Attention: Reports and Documents 
Section, Washington 25, D. C. 


Copy No. 

176 


This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 




SUMMARY TECHNICAL REPORT OF THE 
APPLIED MATHEMATICS PANEL, NDRC 

VOLUME 2 


ANALYTICAL STUDIES IN 
AERIAL WARFARE 


OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

APPLIED MATHEMATICS PANEL 
WARREN WEAVER, CHIEF 


WASHINGTON, D. C1946 




NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 
Roger Adams Army Representative 1 

Frank B. Jewett Navy Representative 2 

Karl T. Compton Commissioner of Patents 3 

Irvin Stewart, Executive Secretary 


1 Army representatives in order of service: 


2 Navy representatives in order of service: 


Maj. Gen. G. V. Strong 
Maj. Gen. R. C. Moore 
Maj. Gen. C. C. Williams 
Brig. Gen. W. A. Wood, Jr. 


Col. L. A. Denson 
Col. P. R. Faymonville 
Brig. Gen. E. A. Regnier 
Col. M. M. Irvine 


Col. E. A. Routheau 


Rear Adm. H. G. Bowen Rear Adm. J. A. Furer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore H. A. Schade 
3 Commissioners of Patents in order of service: 

Conway P. Coe Casper W. Ooms 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities of 
warfare, together with contract facilities for carrying out 
these projects and programs, and (2) to administer the tech¬ 
nical and scientific work of the contracts. More specifically, 
NDRC functioned by initiating research projects on re¬ 
quests from the Army or the Navy, or on requests from an 
allied government transmitted through the Liaison Office 
of OSRD, or on its own considered initiative as a result of 
the experience of its members. Proposals prepared by the 
Division, Panel, or Committee for research contracts for 
performance of the work involved in such projects were 
first reviewed by NDRC, and if approved, recommended to 
the Director of OSRD. Upon approval of a proposal by the 
Director, a contract permitting maximum flexibility of 
scientific effort was arranged. The business aspects of the 
contract, including such matters as materials, clearances, 
vouchers, patents, priorities, legal matters, and administra¬ 
tion of patent matters were handled by the Executive Sec¬ 
retary of OSRD. 

Originally NDRC administered its work through five 
divisions, each headed by one of the NDRC members. 
These were: 

Division A — Armor and Ordnance 

Division B — Bombs, Fuels, Gases, & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three ad¬ 
ministrative divisions, panels, or committees were created, 
each with a chief selected on the basis of his outstanding 
work in the particular field. The NDRC members then be¬ 
came a reviewing and advisory group to the Director of 
OSRD. The final organization was as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 


IV 


CONFIDENTIAL 



NDRC FOREWORD 


As events of the years preceding 1940 revealed more 
-L\. and more clearly the seriousness of the world 
situation, many scientists in this country came to 
realize the need of organizing scientific research for 
service in a national emergency. Recommendations 
which they made to the White House were given care¬ 
ful and sympathetic attention, and as a result the 
National Defense Research Committee [NDRC] was 
formed by Executive Order of the President in the 
summer of 1940. The members of NDRC, appointed 
by the President, were instructed to supplement the 
work of the Army and the Navy in the development 
of the instrumentalities of war. A year later, upon the 
establishment of the Office of Scientific Research and 
Development [OSRD], NDRC became one of its 
units. 

The Summary Technical Report of NDRC is a con¬ 
scientious effort on the part of NDRC to summarize 
and evaluate its work and to present it in a useful and 
permanent form. It comprises some seventy volumes 
broken into groups corresponding to the NDRC 
Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s re¬ 
port contains a summary of the report, stating the 
problems presented and the philosophy of attacking 
them, and summarizing the results of the research, de¬ 
velopment, and training activities undertaken. Some 
volumes may be “state of the art” treatises covering 
subjects to which various research groups have con¬ 
tributed information. Others may contain descrip¬ 
tions of devices developed in the laboratories. A mas¬ 
ter index of all these divisional, panel, and committee 
reports which together constitute the Summary Tech¬ 
nical Report of NDRC is contained in a separate vol¬ 
ume, which also includes the index of a microfilm 
record of pertinent technical laboratory reports and 
reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of suffi¬ 
cient popular interest that it was found desirable to 
report them in the form of monographs, such as the 
series on radar by Division 14 and the monograph on 
sampling inspection by the Applied Mathematics 
Panel. Since the material treated in them is not dupli¬ 


cated in the Summary Technical Report of NDRC, 
the monographs are an important part of the story of 
these aspects of NDRC research. 

In contrast to the information on radar, which is of 
widespread interest and much of which is released to 
the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of Divi¬ 
sion 6 is found almost entirely in its Summary Tech¬ 
nical Report, which runs to over twenty volumes. 
The extent of the work of a division cannot therefore 
be judged solely by the number of volumes devoted 
to it in the Summary Technical Report of NDRC: 
account must be taken of the monographs and avail¬ 
able reports published elsewhere. 

Perhaps the highest tribute which could have been 
paid to the role of mathematicians in World War II 
was the complete lack of astonishment which greeted 
their contributions. To the Applied Mathematics 
Panel of NDRC came urgent, varied, and formidable 
requests from every other group in NDRC and every 
military service. As expected, these requests were 
met; and, also as expected, the results were found 
invaluable in every phase of warfare from defense 
against enemy attack to the design of new weapons, 
recommendations for their use, predictions of their 
usefulness, and analysis of their effects. 

To meet such obligations, the Applied Mathematics 
Panel under the leadership of Warren Weaver, to¬ 
gether with members of its staff and of its contractors’ 
staffs, made available the services of a group of men 
who were not merely able, competent mathematicians 
but also loyal, devoted Americans cooperating unself¬ 
ishly in the defense of their country. The Summary 
Technical Report of the Applied Mathematics Panel, 
prepared under the direction of the Panel Chief and 
authorized by him for publication, is a record of their 
accomplishments and a testimonial to their scientific 
integrity. They deserve the grateful appreciation of 
the Nation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. Conant, Chairman 
National Defense Research Committee 


CONFIDENTIAL 


v 





































































































































































































































































FOREWORD 


hen the National Defense Research Com¬ 
mittee was reorganized at the end of 1942, it was 
decided to set up a new organization, called the Ap¬ 
plied Mathematics Panel [AMP], in order to bring 
mathematicians as a group more effectively into the 
work being carried on by scientists in support of the 
Nation’s war effort. At the time of the original ap¬ 
pointment of the National Defense Research Commit¬ 
tee by President Roosevelt, no mathematicians were 
included on the Committee, and it was not until the 
NDRC had been operating for more than a year that 
the need of a separate division devoted to applied 
mathematics was recognized. Although many of the 
operating Divisions of NDRC had set up mathe¬ 
matical groups to handle their own analytical prob¬ 
lems, it was intended that the new Applied Mathe¬ 
matics Panel should supplement such groups and 
should furnish mathematical advice and service to all 
Divisions of the NDRC, carrying out requested 
mathematical analyses and remaining available as 
consultants after the original analyses had been com¬ 
pleted. The Panel was organized too late to make 
possible a fully definitive trial of the success of this 
type of organization. That mathematics has a funda¬ 
mental role to play in the science of warfare, I am 
sure; I have set forth some of the considerations which 
seem to be relevant and important in the last chapter 
of Volume 2 of the AMP Summary Technical Report. 

The actual development of wartime scientific work 
proved to be such that the Applied Mathematics 
Panel has not only been called upon for assistance by 
NDRC Divisions but has also directly assisted many 
branches of the Army and Navy. Indeed, at the con¬ 
clusion of hostilities, when approximately two hun¬ 
dred studies had been undertaken by the Panel, 
roughly one-half of these represented direct requests 
from the Armed Services. Furthermore, the consult¬ 
ing activities, growing out of studies originally under¬ 
taken to answer specific questions, turned out to be 
considerably more extensive and significant than was 
originally anticipated. I think that the importance of 
this phase of the work cannot be too strongly em¬ 
phasized. But no account of such general consulting 
activities is given here, this report being restricted to 
the formally constituted studies. 

The analytical work under AMP studies was 
carried on by mathematicians associated in groups at 
various universities and operating under OSRD con¬ 
tracts administered by the Panel. To the men who 
served as technical representatives of the universities 
under these contracts, and to the technical aides who 


assisted the Chief in the administration of the Panel’s 
scientific work, the Panel owes a large measure of 
whatever success it achieved. These men combined 
outstanding scientific competence with energy, re¬ 
sourcefulness, and a selfless willingness to devote their 
own efforts, as well as the efforts of their staffs, to the 
solution of other people’s problems. The general plans 
for the Panel’s activities were based upon the counsel 
of a group of eminent mathematicians, formally 
labeled the Committee Advisory to the Scientific Officer. 
This group, meeting every week, and consisting of 
R. Courant, G. C. Evans, T. C. Fry, L. M. Graves, 
H. M. Morse, 0. Veblen, and S. S. Wilks, had 
responsibility for the preliminary examination of re¬ 
quests which reached the Panel and for decisions on 
overall policy. The Chief relied heavily on their ad¬ 
vice which, to a large extent, determined the effective¬ 
ness of the Panel’s activities. 

As the work of NDRC developed, the Panel was 
called upon for assistance by all of NDRC’s nineteen 
Divisions. It is not, therefore, surprising that the 
scope of the Panel’s activities covers a wide range, 
falling into four broad, though somewhat over¬ 
lapping, categories: 

1. Mathematical studies based upon certain classical 
fields of applied mathematics, such as classical me¬ 
chanics and the dynamics of rigid bodies, the theory 
of elasticity and plasticity, fluid dynamics, electro¬ 
dynamics, and thermodynamics. 

2. Analytical studies in aerial warfare , including 
assessment of the performance of sights and antiair¬ 
craft fire control equipment; studies relating to the 
vulnerability of aircraft to plane-to-plane and to anti¬ 
aircraft fire and the optimal defense of the airplane 
against these; and analyses of problems arising from 
the use of rockets in air warfare. 

3. Probability and statistical studies concerned with 
the effectiveness of bombing; various aspects of naval 
warfare, including fire effect analysis and the perform¬ 
ance of torpedoes; the design of experiments; sam¬ 
pling inspection; and analyses of many types of data 
collected by the Armed Services. 

4. Computational services concerned with the eval¬ 
uation of integrals; the construction of tables and 
charts; the development of techniques adapted to the 
solution of special problems; the nature and capa¬ 
bilities of computing equipment. 

The work of the Panel in the first two of these cate¬ 
gories is summarized in Volumes 1 and 2 of the AMP 
Summary Technical Report. Volume 3, together with 

vii 



CONFIDENTIAL 


FOREWORD 


viii 


two monographs a which the Panel has prepared deal¬ 
ing with sampling inspection and techniques of 
statistical analysis, provides a summary of the work 
in the third category. The fourth class of activities 
has been reported in AMP Note 25, Description of 
Mathematical Tables Computed under the auspices of 
the Applied Mathematics Panel , NDRC; in AMP Note 
26, Report on Numerical Methods Employed by the 
Mathematical Tables Project; and in the reports pub¬ 
lished by the Panel under AMP Study 171, Survey of 
Computing Machines. No attempt has been made to 
report on work which will shortly be published as 
articles in scientific journals or on results which are 
deemed too special to be of continuing interest. 

The preparation of this Summary Technical Re- 

a Sampling Inspection and Techniques of Statistical Analysis, 
published by the McGraw-Hill Book Co., Inc. 


port was undertaken after the end of World War II, 
at a time when the members of the Panel's staff and 
of the contract groups were eager to return to their 
peacetime careers. Thus the preparation of these 
three volumes, solely for the purpose of recording for 
the Services, in easily accessible form, the scientific 
results of the Panel's activities, was achieved at real 
personal sacrifice. I am greatly indebted to the 
authors of the several parts of these volumes and to 
the Editorial Committee, consisting of Mina Rees, 
I. S. Sokolnikoff, and S. S. Wilks, for the admirable 
job they have done in bringing together, under high 
pressure, a summary of the principal scientific ac¬ 
complishments of the Panel. 

Warren Weaver 
Chief, Applied Mathematics Panel 


CONFIDENTIAL 



PREFACE 


I 


T his volume furnishes a summary of those ac¬ 
tivities of the Applied Mathematics Panel which 
were concerned with air-to-air, ground-to-air, or air- 
to-ground warfare exclusive of bombing, the majority 
of studies having to do with the design and use of fire 
control equipment. Although much of this work was 
undertaken at the request of that part of the NDRC 
which was in charge of research and development in 
the whole field of fire control, namely, Division 7, 
many requests were received from other Divisions of 
NDRC, and many came directly from the Army and 
Navy and from the Joint Army-Navy-NDRC Air¬ 
borne Fire Control Committee. Rarely did these re¬ 
quests involve studies which could be made in time 
to influence design — a situation which arose at least 
partially from the imperative need to obtain results 
which could be used in World War II — so that most 
of the studies were concerned either with the im¬ 
provement of the theoretical accuracy of equipment 
by suitable changes in design, or with the best use of 
existing equipment. Nevertheless, in the attempt to 
answer specific questions, basic results were derived 
which should be of continuing interest. In the account 
here given, it is the basic theory which is emphasized, 
although a brief account of many specific results is 
included. Because of the diversity of the sources from 
which requests were received, and the varying re¬ 
quirements of the requesting agencies, it has been 
necessary, particularly in Part III (Antiaircraft 
Analysis), to report on studies which are somewhat 
disconnected. In both chapters of this part, a general 
introductory discussion of the nature of the problem 
is given which will, it is hoped, serve as a background 
for the more detailed treatment which follows. 

By far the most extensive analyses carried on by 
the Panel in aerial warfare were concerned with air- 
to-air gunnery, the work in one contract being for 
several years devoted almost exclusively to this 
phase of warfare, while several other contracts were 
concerned with it for shorter periods of time and, in 
some cases, as incidental aspects of their work. The 
Panel was fortunate in having as the Director of the 
Applied Mathematics Group at Columbia (the con¬ 
tract with the most extensive and longest experience 
in this field) Saunders MacLane, who combined with 
outstanding mathematical competence the energy 
and personal effectiveness which kept the Group 
activities in close touch with Army and Navy needs. 
The Naval Ordnance Development Award has been 


conferred on the Group for distinguished service to 
the research and development of naval ordnance, and 
in particular for its contribution to the development 
of gunsights Mark 18 and 23. At Northwestern, where 
a Panel group headed by Walter Leighton worked in 
close touch with Division 7 and the Patuxent Naval 
Air Station, the principal concern was with the 
development of methods for the experimental assess¬ 
ment of fire control systems for aerial gunnery, with 
emphasis on camera techniques. 

The report on Aerial Gunnery is presented as Part 
I of this volume, and in it a serious attempt has been 
made to include an account not only of the Panel's 
work in the field, but of all available literature re¬ 
gardless of source. The Panel’s assigned emphasis on 
the analytical aspects of the subject is reflected in the 
emphasis on Panel studies which is bibliographically 
evident. The reader is referred also to an account of 
aerial gunnery which has been given by Saunders 
MacLane a in which he surveys rapidly, informally, 
and in a nontechnical manner the material covered 
in Part I of this report. Throughout the MacLane 
paper, critical and specific statements based on wide 
experience will be found of the way gunnery research 
was handled during World War II. The present ac¬ 
count does not move in that direction. It attempts 
rather to present a certain approximation to the 
“state-of-the-art.” The body of the report tries to 
indoctrinate new technical workers in aerial gunnery; 
the introductions and summaries of the several 
chapters may well interest a more general reader. 
The field of the account is limited by the omission of 
(1) engineering details of the various elements of fire 
control systems, (2) the maintenance of those 
systems, (3) the training required of aerial gunners, 
(4) the results of laboratory and airborne experi¬ 
mental programs, and (5) the analysis of gunnery in 
combat. In the positive direction, the account surveys 
underlying ballistic, deflection, and aerodynamic 
theory, and considers the ways in which fire control 
systems attempt to solve the problem put to them 
and the errors these systems make in that effort. 
Although this is a summary report, enough technical 
detail is supplied to make the report, in large measure, 
self-contained. In the selection, translation, and 
grouping of material from many sources, the author 
has performed a most difficult task with extraordi- 

a Aerial Gunnery Problems , AMG-C Paper 491, Columbia 
University, August 31, 1945. 


CONFIDENTIAL 


IX 


X 


PREFACE 


nary clarity and effectiveness. Because so many of 
the papers which are reviewed were written in reply 
to specific questions which needed quick answers, the 
job of presenting a unified picture of the work was 
particularly difficult. The account reflects the unique 
experience of its author, E. W. Paxson, with Army, 
Navy, NDRC, and British as well as German 
activities. 

In Part I of this volume, the behavior of weapons 
and their control mechanisms is studied rather than 
the tactical employment and strategic consequences 
of the use of those weapons. In Part IV, a brief report 
is given of two studies also dealing with air-to-air 
warfare which came closer to having general tactical 
or strategic scope than did most other AMP studies. 
Groups at New Mexico under E. J. Workman, at 
Mount Wilson Observatory under W. S. Adams, and 
at Princeton under M. M. Flood carried on the ex¬ 
tensive researches reported here which were con¬ 
cerned with the best tactical use of the B-29, while 
the Statistical Research Group at Columbia, whose 
director of research was W. Allen Wallis, was respon¬ 
sible for the study of alternative fighter plane arma¬ 
ment. In addition to reporting on these two studies, 
Part IV gives a discussion of a general theory of air 
warfare and of some of the contributions which 
mathematics can make to the broad field of national 
defense. It was P. M. S. Blackett, a British pioneer in 
the field of operational research, who pointed out that 
it is the study of how and why weapons perform and 
how they may be improved that is amenable to the 
usual approach of the physical sciences, whereas the 
study of how tactical procedures may be improved 
and the determination of the costs in resources of war 
to modify strategic concepts require statistical and 
variational methods. In Part IV some indication is 
given by the Chief of the Applied Mathematics Panel 
of how certain activities of the Panel and of other 
agencies relate to a scheme for a broader analytical 
approach to the problems of air warfare and of war¬ 
fare in general. 

Part II is concerned with the sighting methods 
which are feasible for airborne rockets. Although an 
important part of the Panel's work in rocketry con¬ 
sisted of conferences on sighting methods and re¬ 
lated problems with various groups and in solving 
problems connected with special sights, no account 
is here given of the special results obtained except 
for a brief mention in the introduction to Chapter 9 
where bibliographical source material is indicated. 
The account presented in Part II is concerned only 


with that part of the Panel’s work in the field which 
is deemed to be of importance for the future. No 
author is indicated for Part II, since the chapter 
represents almost wholly a digest prepared in the 
Panel office of work which originated with Hassler 
Whitney, who was in charge of the Panel’s rocketry 
program. Whitney served as a member of the 
Applied Mathematics Group at Columbia; he not 
only integrated the work carried on at Columbia and 
at Northwestern for the Panel in the general field 
of fire control for airborne rockets, but maintained 
effective and continuous liaison with the work of 
Division 7 in this field and with the activities at many 
Army and Navy establishments, particularly the 
Naval Ordnance Test Station at Inyokern, the 
Lukas-Harold Corporation, the Dover Army Air 
Base, Wright Field Armament Laboratory, the 
Naval Bureau of Ordnance, and the British Air 
Commission. 

The first chapter of Part III is concerned princi¬ 
pally with an account of results obtained by the 
Panel as an outcome of various requests for the 
analysis of antiaircraft equipment. Most of this work 
was done either by the Applied Mathematics Group 
at Columbia or by the Columbia Statistical Research 
Group. The second chapter of Part III is concerned 
with fragmentation and damage studies. In this field 
the basic theory was developed by the British, who 
also obtained important experimental results con¬ 
cerning the fragmentation characteristics of shells. 
The Panel used British, Army, Navy, and OSRD 
reports as the source of its experimental information. 
Its own contribution was in developing analytical and 
computing procedures which were feasible in point 
of time and in applying these procedures to selected 
examples. The greater part of the work performed 
for the Panel in this field was carried out by the 
Statistical Research Group at Columbia where 
Milton Friedman, Associate Director of the Group, 
became an expert in the field and served as consult¬ 
ant to the many Army, Navy, and OSRD groups 
which had frequent occasion to seek his assistance. 
One major report in this field was prepared by the 
Applied Mathematics Group at Brown University. 

Because so much of the work reported in this 
volume was concerned with equipment developed by 
Division 7, the reader interested in this subject will 
do well to consult the Summary Technical Report of 
Division 7. For a discussion of the characteristics and 
present stage of development of airborne radar fire 
control systems and related problems such as the 


CONFIDENTIAL 



PREFACE 


XI 


coordination of radar and computers, the reader 
should refer to MARS, Volume 2 of the Summary 
Technical Report of Division 14, Chapters 17 to 22. 

It has been the aim of the authors of this volume 
to present the material in such a way that no prior 
knowledge of specific technical matters is presupposed 
on the part of the reader. The reader is expected to 
have a background in mathematics and physics 
ordinarily possessed by a person with a bachelor’s 
degree in engineering. 


The bibliographies to the various parts of this vol¬ 
ume give some indication of the scope of the material 
which the authors have examined in the writing of 
this book. They are to be congratulated for having 
prepared accounts which, in spite of the great di¬ 
versity in the work surveyed and the brief time 
available for the preparation of the report, should 
prove useful to future workers in the field. 

Mina Rees 
Editor 


CONFIDENTIAL 







CONTENTS 


CHAPTER PAGE 

Summary . 1 

PART I 

ANALYTICAL ASPECTS OF AERIAL GUNNERY 

1 Aeroballistics. 9 

2 Deflection Theory.22 

3 Pursuit Curves. 30 

4 Own-Speed Sights.45 

5 Lead Computing Sights.57 

6 Central Station Fire Control.82 

7 Analytical Aspects of Airborne Experimental Programs . 94 

8 New Developments.107 

PART II 
ROCKETRY 

9 Fire Control for Airborne Rockets.125 

PART III 

ANTIAIRCRAFT ANALYSIS 

10 Studies of Antiaircraft Equipment.145 

11 The Risk to Aircraft from High-Explosive Projectiles . . 167 

PART IV 
GENERAL 

12 Comments on a General Theory of Air Warfare . . . . 197 

Appendix A.221 

Appendix B.223 

Bibliography .227 

OSRD Appointees.239 

Contract Numbers.w.240 

Project Numbers.242 

Index.245 


CONFIDENTIAL 

























• *'**'•»»»»»**«•* 








•* 


* 


» 


» 




* • • 


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**»• 




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SUMMARY 


I N this Summary Technical Report of the Applied 
Mathematics Panel, a resume is given of the prin¬ 
cipal scientific accomplishments of the Panel from its 
beginning in 1943 until the conclusion of hostilities. 
The activities here reported cover a wide range, deal¬ 
ing as they do with studies undertaken at the request 
of each of the nineteen Divisions of NDRC and of 
many branches of the Army and Navy. For the pur¬ 
pose of this report, that portion of the Panel’s work 
which deals with specific military problems has been 
divided into three parts: Volume 1 , Mathematical 
Studies Relating to Military Physical Research; 
Volume 2, Analytical Studies in Aerial Warfare; and 
Volume 3, Probability and Statistical Studies in War¬ 
fare Analysis. In addition to reporting on specific 
military problems, Volume 1 also indicates directions 
in which certain of the theories of fluid dynamics have 
been extended under AMP auspices as an aid in the 
planning and interpretation of military experiments, 
and in understanding the operation of enemy 
weapons. These three volumes contain no account of 
the new developments in statistical methods which 
have already been partially reported in a published 
article 1 and a published book 2 on sequential analysis, 
nor of certain important new applications of statisti¬ 
cal theory which grew out of the Panel’s attempt to 
solve problems presented to it by the Services. These 
latter are reported in two published monographs, 
Sampling Inspection and Techniques of Statistical 
Analysis (published by McGraw-Hill), which have 
been prepared under Panel auspices and which form 
part of the Panel’s report of its technical activities. 

Most AMP studies were concerned with the im¬ 
provement of the theoretical accuracy of equipment 
by suitable changes in design; or with the develop¬ 
ment of basic theory, particularly in the field of fluid 
dynamics; or with the best use of existing equipment, 
particularly in fields like bombing and the barrage 
use of rockets. Two studies carried out under AMP 
auspices come closer to having general tactical or 
strategic scope than do most of the other work. I 
have myself given an account of these two studies in 
Part IV of Volume 2, where I have also set forth 
some incomplete and preliminary ideas of what a 
general analytical theory of air warfare could and 
should comprise and some arguments for and against 
attempting to construct and use such a theory. I have 
there indicated how certain activities of the Applied 
Mathematics Panel and of other agencies relate to a 


scheme for a broad approach to the problems of air 
warfare and of warfare in general, and I have pointed 
out some of the contributions which mathematics can 
make to the field of national defense. 

That part of the Panel’s work which may be 
roughly described as classical applied mathematics is 
presented in Volume 1. Certain phases of this subject 
were developed under Panel auspices and adapted 
to problems of military interest, the principal 
emphasis being on problems of primary concern to 
the Navy. 

In the early stages of the war, certain acoustic 
equipment employed in submarine detection by echo 
ranging used a “dome” — a streamlined convex shell 
filled with water or other liquid, such as oil. The 
presence of these domes caused interference with the 
directional pattern sent out from the projector, and 
in some of the equipment the disturbance was ex¬ 
tremely serious. The Panel was asked to study the 
situation and to suggest changes in the domes which 
would minimize the disturbances. Practical con¬ 
clusions were reached regarding desirable materials 
and design. It was found desirable for practical 
reasons to use thin shells reinforced by stiffening 
elements such as ribs and rods rather than to achieve 
strength by general thickness. Difficulties arising in 
direction finding due to annoying reflections were 
also analyzed, and suggestions were made for im¬ 
proving conditions, for example, by corrugations on 
the inner surface of the side walls of the domes. This 
dome study was one aspect of the work in wave 
propagation with which the Panel was concerned. 
There were others. For example, an investigation 
was made of the scattering of electromagnetic waves 
by spherical objects to assist in the analysis of smokes 
and fogs. A study of somewhat similar mathematical 
character (but dealing with electromagnetic disturb¬ 
ances rather than actual mechanical waves in a 
liquid) was undertaken at the request of the Fire 
Control Division (Division 7, NDRC), which had 
under development a predictor, the T-28, intended 
for use with the 40-mm gun. The computing mechan¬ 
ism used by this predictor included a sphere on which 
were placed electrical windings in such a way that 
the resulting field was one which corresponded to one 
simple dipole at the center of the sphere. Although 
the theoretical way in which the winding should be 
distributed on the surface of this sphere was well 
known, it was necessary as a practical matter to 


By Warren Weaver. 


CONFIDENTIAL 


1 



2 


SUMMARY 


substitute a winding in which the turns were located 
in grooves on the sphere. The formulas resulting from 
the PaneFs study of this problem form a basis for 
practical applications which include ammeters, 
galvanometers, and direction finders. This mathe¬ 
matical study was of critical importance for the fire 
control instrument in question, for without it, it was 
impossible to obtain useful accuracy in the spherical 
“electromagnetic resolver” which carried out the 
essential steps in the target predicting process. 

The PaneFs work in gas dynamics, mechanics, and 
underwater ballistics is also reported in this first 
volume. The PaneFs work in gas dynamics was prin¬ 
cipally concerned with the theory of explosions in the 
air and under water, and with certain aspects of jet 
and rocket theory. New developments were made in 
the study of shock fronts, associated with violent 
disturbances of the sort which result from explosions. 
An interesting and significant aspect of the work was 
concerned with Mach phenomena which frequently 
play a practical role in determining the destructive 
effects of shocks. For example, the advantages of air¬ 
bursting large blast bombs were suggested by a con¬ 
sideration of Mach waves. A request from the Bureau 
of Aeronautics for assistance in the design of nozzles 
for jet motors to be used for assisted take-off gave 
rise to an extended study of gas flow in nozzles and 
supersonic gas jets. As a result, suggestions were made 
not only for the design of nozzles for jet-assisted take¬ 
off, but also for “perfect” exhaust nozzles and com¬ 
pressors (of use in supersonic wind tunnels) and for 
various instruments to aid in rocket development and 
experimentation. The jet propulsion studies were re¬ 
lated to Army and Navy interest in intermittent jet 
motors of the V-l type. Jet propulsion under water 
was also studied, with results which should prove use¬ 
ful as a guide to experiment in this field where experi¬ 
mentation has thus far not reached the stage where 
the theoretical results can be fully put to test. 

The problems in mechanics fall under two general 
headings: (1) those involving the mechanics of par¬ 
ticles and rigid bodies and (2) those involving the 
mechanics of a continuum. For example, a study in 
the second category sought possible explanations of 
the break-up in cylindrical powder grains in the 
43^-in. rocket to explain difficulties which were being 
encountered at the Allegany Ballistics Laboratory, 
and an experimental program was outlined for the 
testing of the most probable theories. One of the most 
interesting of the mechanical studies concerned the 
so-called spring hammer box used by the U. S. Navy 


in acoustic mine warfare. The dependence of the 
operation of this device on various physical param¬ 
eters (for example, the mass of the hammer) was 
analyzed with the aid of a simple mechanical model, 
and of an electrical analog. Another problem of this 
type studied the dynamics of the gun equilibrator, or 
balancing system, when an Army gun was mounted 
on board a ship. The pitching and rolling of the ship 
naturally introduced special difficulties. 

In the section on underwater ballistics, the prob¬ 
lems involved are classified according to the various 
phases in the motion of the projectile: the impact 
phase, the development of the cavity, and the under¬ 
water trajectory. During the impact phase, forces act 
which are important partly because of their possible 
effects on the nose structure and mechanism of the 
projectile, partly because of their influence in deter¬ 
mining the projectile’s subsequent motion. It is dur¬ 
ing the impact phase that the greatest deceleration 
occurs. The theoretical analysis involves, among 
many other considerations, the direction of entry 
(vertical or oblique), and the shape of the projectile. 
Save when the speed of a missile is slow, its entry is 
accompanied by the formation of a cavity which be¬ 
comes sealed behind the projectile and accompanies 
it to a greater or less extent during its underwater 
motion, influencing that motion in an important way. 
The underwater trajectory itself presents problems 
of great complexity. Frequently, slight changes in 
values of the parameters which determine the motion 
will cause a complete change in the type of motion. 
A mathematical discrimination among the several 
types of motion is made, part of the distinction de¬ 
pending on such things as the position of the center 
of gravity of the missile, the ratio of its length to its 
diameter, its density, its radius of gyration, and the 
manner of its entry. Throughout this treatment, an 
attempt has been made to integrate into a single re¬ 
port the results which have been obtained by the 
many agencies concerned with the several phases of 
the problem and thus to assist the theoretical and 
experimental studies which must be carried forward 
in future attempts to understand this difficult array 
of problems. 

Many of the studies reported in Volume 2, as well 
as those contained in Volume 3, involve probability 
considerations, a field which is notoriously tricky and 
within which “common sense” is often quite helpless. 
For example, what is the optimum mixture of armor¬ 
piercing and incendiary ammunition for the rear guns 
of a bomber? Specifications often designate such 


CONFIDENTIAL 



SUMMARY 


3 


mixtures as five AP to two incendiary (we are 
neglecting tracers here). Why? The somewhat strik¬ 
ing, and by no means obvious, fact is that, given any 
fixed type of target, it is better to have either all AP 
or all incendiary, depending on the nature of the 
target. The justification for any other intermediate 
mixture should be based on knowledge of the relative 
probability of encountering different targets, certain 
of which would be more vulnerable to AP and others 
more vulnerable to incendiary. This conclusion was 
reached as an incidental result of a study which was 
concerned with alternative fighter-plane armament 
and which arose out of the enthusiasm of a few per¬ 
sons associated with the Panel for two papers at¬ 
tributable to L. B. C. Cunningham, Chief of the Air 
Warfare Analysis Section in England, and his as¬ 
sociates. Another study concerned with the practical 
effectiveness of equipment grew out of a request to 
NDRC from Headquarters, AAF, asking for collabo¬ 
ration with the AAF “in determining the most ef¬ 
fective tactical application of the B-29 airplane.” 
The results of this study, obtained on the basis of 
large-scale experiments in New Mexico and small- 
scale optical experiments by the Mt. Wilson Ob¬ 
servatory staff at Pasadena, were concerned princi¬ 
pally with the defensive strength of single B-29’s and 
of squadrons of B-29’s against fighter attack and the 
effectiveness of fighters against B-29’s. One indirect 
result of the optical studies was a set of moving 
pictures showing the fire power variation of forma¬ 
tions as a fighter circles about them. Concerning such 
pictures the President of the Army Air Forces Board 
remarked that he “believed these motion pictures gave 
the best idea to air men as to the relative effect of fire 
power about a formation yet presented.” Certain of 
these pictures were flown to the Mariannas and 
viewed by General LeMay and by many gunnery 
officers at the front. 

These two studies are reported in the last part of 
Volume 2. The first three parts of this volume report 
on special and detailed problems which arise when 
shots are fired against targets moving in the air or on 
the ground. The problem of shooting from an aircraft 
in motion against an enemy aircraft or against a 
ground target in motion and the problem of shooting 
from the ground or from a naval craft against an 
enemy aircraft all involve a number of considerations. 

1. Whenever the target is in motion, its position at 
the instant of firing is different from its position at 
impact, if impact occurs. For an effective shot, the 
motion of the target during the time of flight of the 


bullet or rocket or shell must therefore be predicted, 
at least approximately. The special character of this 
problem for the special cases which have come under 
the Panel’s study are discussed for air-to-air warfare 
in Part I, for rocket fire from the air in Part II, and 
for ground or ship based antiaircraft fire in Part III. 

2. When one’s own ship is in motion, the apparent 
motion of the target is affected. 

3. There are oscillations in aim as the gunner at¬ 
tempts to point continuously at the target. These 
oscillations are greater in air-to-air and in ship-to-air 
than in ground-to-air gunnery because of the vibra¬ 
tions, rotations, and bumpy motions of one’s own 
ship. 

4. There is the effect of gravity on the bullet. In 
air-to-air gunnery, for the short ranges used in World 
War II, this was of minor importance, but for rocket 
fire it introduced very considerable complications. 

5. The resistance of the air varies with the altitude. 
Thus, at 22,000 feet above sea level the air is half as 
dense as it is at sea level. This will affect the average 
speed of a bullet, hence its time of flight, and hence 
the prediction referred to above. 

A large part of Volume 2 is devoted to problems 
connected with so-called flexible gunnery, i.e., with 
the aiming of those guns, carried on aircraft, which 
can be pointed in various directions with respect to 
the aircraft (as contrasted with fixed guns in the 
wings or nose, which are aimed only by movement 
of the aircraft). In January 1944, Brigadier General 
Robert W. Harper, AC/AS (Training), wrote in a 
letter to Dr. Vannevar Bush, Director of OSRD, that 
“the problems connected with flexible gunnery are 
probably the most critical being faced by the Air 
Forces to-day. It would be difficult to overstate the 
importance of this work or the urgency of the need; 
the defense of our bomber formations against fighter 
interception is a matter which demands increasing 
coordinated expert attention.” This situation arose 
because of the inadequate training and inadequate 
deflection rules given to the gunners who had to 
handle ring sights in bombers. The “relative speed” 
and “apparent motion” rules currently taught were 
not thoroughly learned by the gunners and in many 
cases were by no means adequate when they were 
properly applied. There were well authenticated 
cases of gunners who “led” the attacking fighters in 
a direction exactly opposite to that of the true lead! 

The immediate proposal contained in General 
Harper’s letter was that the Applied Mathematics 


CONFIDENTIAL 



4 


SUMMARY 


Panel should recruit and train competent mathe¬ 
maticians who had the “ versatility, practicality, and 
personal adaptability requisite for successful service 
in the field;” it was planned that these men, after 
two months’ training in this country, would be as¬ 
signed to the Operations Research Sections in the 
various theaters to devote their attention to aerial 
flexible gunnery problems. The Panel was in a posi¬ 
tion to carry out this program because it had already 
been drawn into studies of rules for flexible gunnery 
training and because it had access to many of the 
ablest young mathematicians in the country. The 
assignment was completed promptly, and, as a 
partial result of this undertaking, the Panel found 
itself even more closely in touch with the Operations 
Analysis Division of the AAF (with which it had 
already established cordial working relations) and 
with the AAF Central School for Flexible Gunnery. 
Around this interest and the interest of the Army, 
the Navy, Division 7, and Division 14 in the im¬ 
provement in the effectiveness of guns as well as 
gunnery, grew up a very considerable body of knowl¬ 
edge and experience which is reported in Part I of 
Volume 2. Here an attempt is made to bring together 
into a single account the state of the art of air-to-air 
gunnery, not only as that has been affected by the 
work of the Applied Mathematics Panel, but as it 
has reflected the activities of agencies in this country 
and abroad. The topics discussed are: 

1. The motion of a projectile from an airborne gun, 
constituting that branch of exterior ballistics which 
is called aeroballistics. 

2. A mathematical theory of deflection shooting con¬ 
sidered first for the case of a target moving at con¬ 
stant speed on a straight line which lies in a plane 
with the gun-mount velocity vector; second, for a 
target which moves in a curved path; and third, for 
the case where mount and target move in arbitrary 
space paths. 

3. Pursuit curve theory. Pursuit curves were im¬ 
portant in World War II, since the standard fighter 
employed a heavy battery of guns so fixed in the air¬ 
craft as to fire sensibly in the direction of flight. Thus 
it was necessary to fly on such a correctly banked 
turn that a correct and changing aiming allowance 
was continuously made. This pursuit curve theory is 
also of importance in the study of guided missiles 
which continuously change direction under radio, 
acoustic, or optical guidance unwillingly supplied by 
the target. 


4. The design and characteristics of own-speed 
sights which were introduced as devices designed for 
use against the special case of pursuit curve attack 
on a defending bomber. Simple charts which might 
be used in the air are given, based on optimum rules 
for determining deflection against an aerodynamic 
pursuit curve. 

5. Lead computing sights which do not assume that 
the fighter is coming in on a pursuit curve but which 
basically assume that the target’s track relative to 
the gun mount is essentially straight over the time 
of flight of the bullet. The mechanical sights of the 
Sperry series are considered in some detail. 

6. The basic theory of a central station fire control 
system. 

7. The analytical aspects of experimental programs 
for testing airborne fire control equipment. It is recog¬ 
nized that field tests, laboratory tests, and theoretical 
analyses all have an important place in such a pro¬ 
gram. Instrumentation for tests, reduction of data, 
measures of effectiveness, and optimum dispersion 
are discussed. 

8. New developments, such as stabilization and the 
use of radar. 

The second part of Volume 2 is devoted largely to a 
presentation of the results obtained by the Panel in a 
study intended to determine what sighting methods 
are feasible for airborne rockets. The essential prob¬ 
lems involved in this question have to do with ballistic 
formulas, attack angle and skid, the effect of wind 
and target motion, how these various factors affect 
each proposed sighting method, and how tracking 
affects and is affected by them. 

In Part III of Volume 2 certain special studies of 
antiaircraft equipment which were made under AMP 
auspices are discussed, and a report is given of the 
flak analysis and other fragmentation and damage 
studies carried on by the Panel. This report is con¬ 
cerned with some mathematical problems which 
arise in attempts to estimate the probability of 
damage to an aircraft or group of aircraft from one or 
many shots from heavy antiaircraft guns. Related 
problems arise in air-to-air bombing and in air-to-air 
or ground-to-air rocket fire, but the major part of the 
mathematical analysis so far performed has been 
devoted to problems of flak risk. The emphasis in the 
discussion is on the description of a method for treat¬ 
ing problems of risk, since specific numerical con¬ 
clusions are likely to become obsolete before further 
need for them arises, while the techniques by which 


CONFIDENTIAL 



SUMMARY 


5 


the results were obtained will be useful as long as 
weapons which destroy by means of flying fragments 
are in use. The original experimental information on 
which the Panel computations were based came from 
a variety of sources, principally Army, Navy, OSRD, 
and British reports. The Panel’s chief contribution 
was the development of computational techniques 
which could be carried through before the project be¬ 
came obsolete, the selection of pertinent examples, 
and the applications of the computational techniques 
to the selected examples. Certain applications of the 
underlying theory to time-fuzed and proximity-fuzed 
shells, and to proximity-fuzed rockets are here re¬ 
ported. 

Another major field of effort in the work of the 
Panel is that of Mathematical Statistics, reported in 
Volume 3. A remarkably wide variety of probability 
and statistical investigations was carried out by the 
Panel. These investigations ranged from the develop¬ 
ment of sampling inspection plans in connection with 
procurement of military material to extensive sta¬ 
tistical analyses of combat data. Of the Panel’s 194 
studies, 53 related to problems in probability and 
statistical analysis. 

The work of the Panel in mathematical statistics 
can be grouped into the following major categories: 

1. Bombing accuracy research. 

2. Development of statistical methods in inspection, 
research, and development work. 

3. Development of new fire effect tables and dia¬ 
grams for the Navy. 

4. Miscellaneous studies relating to spread angles 
for torpedo salvos, lead angles for aerial torpedo attacks 
against maneuvering ships, land mine clearance, per¬ 
formance of heatrhoming devices, search problems, veri¬ 
fication of weather forecasting for military purposes, 
procedures for testing sensitivity of explosives, distri¬ 
bution of Japanese balloon landings, etc. 

Of these four main categories of work, category 1 
required by far the greatest amount of energy. This 
activity had its beginning in a fairly small study 
undertaken for the Armament Laboratory, Wright 
Field, on the design of a computer for determining the 
optimum spacing of bombs in a train of bombs 
dropped from a bomber in attacking a given target 
under specified conditions. The study was started in 
1942 under Division 7, NDRC, and was transferred 
to the Panel when the Panel was organized. In pur¬ 
suing this study the group working on it came in 
contact with individuals in more than a dozen Army, 


Navy, and NDRC groups interested in bombing ac¬ 
curacy problems. As the war progressed, an increas¬ 
ing number of requests came from these groups for 
studies of all kinds of accuracy and coverage prob¬ 
lems arising in train bombing, area bombing, pattern 
bombing, guided-missile bombing, incendiary bomb¬ 
ing, and so on. By the end of the war the work in this 
field had grown to the point where the major effort 
of three Panel research groups was being spent on 
nineteen studies dealing with probability and statisti¬ 
cal aspects of bombing problems. 

The methods and results developed in category 2 
are of much broader interest than that associated 
with their wartime applications. During the war, it 
was recognized by the Services that the statistical 
techniques which were developed by the Panel for 
Army and Navy use, on the basis of the new theory 
of sequential analysis, if made generally available to 
industry, would improve the quality of products pro¬ 
duced for the Services. In March 1945, the Quarter¬ 
master General wrote to the War Department liaison 
officer for NDRC a letter containing the following 
statement: 

“By making this information available to Quartermaster 
contractors on an unclassified basis, the material can be 
widely used by these contractors in their own process control 
and the more process quality control contractors use, the 
higher quality the Quartermaster Corps can be assured of ob¬ 
taining from its contractors. For, by and large, the basic cause 
of poor quality is the inability of the manufacturer to realize 
when his process is falling down until he has made a considera¬ 
ble quantity of defective items. . . . With thousands of con¬ 
tractors producing approximately billions of dollars worth of 
equipment each year, even a 1% reduction in defective mer¬ 
chandise would result in a great saving to the Government. 
Based on our experience with sequential sampling in the past 
year, it is the considered opinion of this office that savings of 
this magnitude can be made through wide dissemination of 
sequential sampling procedures.” 

On the basis of this and similar requests, the 
Panel’s work on sequential analysis was declassified, 
and the reports mentioned above were published. The 
Quartermaster Corps reported in October 1945 that 
at least 6,000 separate installations of sequential 
sampling plans had been made and that in the few 
months prior to the end of the war new installations 
were being made at the rate of 500 per month. The 
maximum number of plans in operation simultane¬ 
ously was nearly 4,000. 

Thus extensive use was made by the Army of 
sequential analysis as a basis for sampling inspection. 
It was at the request of several Navy bureaus that 


CONFIDENTIAL 



6 


SUMMARY 


the Panel undertook to assemble a manual setting 
forth procedures to be used not only with sequential 
sampling but also with single and double sampling 
plans. As an extension and expansion of this manual, 
the Panel undertook the preparation of its mono¬ 
graph, Sampling Inspection. The monograph, Tech¬ 
niques of Statistical Analysis , presents a variety of 
statistical methods which have been developed, or 
adapted from more general methods, for dealing with 
various statistical problems which have arisen in 
connection with research and development work. 

The work done in category 3 was of highly special¬ 
ized long-range interest to the Office of the Com¬ 
mander in Chief of the U. S. Fleet. After the work 
had been carried forward under the direction of the 
Panel for nearly two years, arrangements were made 
to transfer and continue the work under a contract, 
effective June 1,1945, between the Navy and Prince¬ 
ton University. During the time this work was under 
the Panel’s direction, a series of nine basic reports 
was submitted to the Navy. None of this work, which 
was only partially completed under the direction of 
the Panel, is reported upon in the Panel’s Summary 
Technical Report. 

Certain of the studies in category 4 are of such 
limited interest that it has been considered neither 
appropriate nor worth while to report upon them 
here. Accounts are given of the work which relates 
to torpedoes, land mine clearance, and the perform¬ 
ance of heat-homing devices. 

An important adjunct of the probability and 
statistical work of the Panel was a statistical con¬ 
sulting service for various Army, Navy, and NDRC 
agencies. Although some of this consulting was done 
in connection with formal AMP studies and projects 
in such a way that the results are adequately reported 
in original Panel reports or the Panel’s Summary 
Technical Report, a large fraction of it was informal 
and the results of it are to be found in reports and 
memoranda of many agencies, particularly Divisions 


2, 5, 8, and 11 of NDRC; Joint Army-Navy Target 
Group, Army Air Forces Board; Proving Ground 
Command, Eglin Field, AAF; Operational Analysis 
Division, Twentieth Air Force, AAF; Combat An¬ 
alysis Unit, Statistical Control, AAF; Office of the 
Quartermaster General; Navy Air Intelligence Group; 
Navy Operational Research Group; and the Guided 
Missile Committee of the Joint Chiefs of Staff. 

Men from several of the Panel’s research groups 
acted as consultants to these various agencies for 
periods ranging from two months to two years. In my 
opinion some of the most useful service which the 
Panel was able to render came about through the 
work of these men in their capacities as consultants; 
the effectiveness of this work increased constantly 
until the end of the war. The work of these men varied 
widely: assistance in setting up sampling inspection 
plans for procurement of materiel, helping in the in¬ 
troduction of a quality control system in rocket pro¬ 
duction, working on designs of experiments for toxic 
gas bombing, testing controlled missiles, cooperation 
in the preparation of an incendiary manual, and 
dozens of other projects. 

I cannot leave the topic of mathematical statistics 
without emphasizing the powerful yet severely prac¬ 
tical role which this relatively young branch of ap¬ 
plied mathematics has played in the work of the 
Panel. The tools of the probabilitist and statistician 
have been applied to an almost unbelievably wide 
array of problems. Probability analysis played a 
fundamental part in a priori investigation of various 
kinds of weapons and tactics studied by the Panel. 
As the war progressed and these weapons and tactics 
were tested at the proving ground and tried out in 
combat, the analysis of the observational data be¬ 
came primarily statistical. The work of the Panel 
surely indicates that the Army and Navy will do well 
in their research, development, and testing of weapons 
and tactics to see to it that the tools of the mathe¬ 
matical statistician are not overlooked. 


CONFIDENTIAL 



PART I 

ANALYTICAL ASPECTS OF AERIAL GUNNERY 


CONFIDENTIAL 





Chapter 1 


AEROBALLISTICS 


l.l INTRODUCTION 

T he discussion of the motion of projectiles fired 
from airborne guns constitutes a modern branch 
of exterior ballistics which may quite properly be 
called aeroballistics. Certain essential points of dif¬ 
ference between classical exterior ballistics and aero¬ 
ballistics are considered below. 

1. The ranges employed in aeroballistics have 
been short compared with the maximum effective 
range of the projectile. During World War II these 
ranges were, in general, no greater than 1,000 yd. 
For such ranges the effects of changes in density, 
temperature, and wind along the trajectory are negli¬ 
gible. Bullet drift may also be neglected. Since the 
projectile velocity does not become subsonic, the 
resistance encountered by the bullet is proportional 
to the three-halves power of the speed, and ballistic 
formulas for the trajectory may therefore be written 
out in closed and compact form. For short ranges, the 
trajectory is relatively flat, so that consideration of 
superelevation to allow for gravity drop is of minor 
importance. 

It can be pointed out that a moving target may be 
hit at long range 210 using a highly arched trajectory, 
as well as at short range with a flat trajectory. But 
the difficulty in predicting the position of the target 
over a long time of flight of the projectile and the 
inaccuracy in positioning an airborne gun rule out 
long-range fire. In addition, the remaining velocity 
would probably be too low to achieve effective dam¬ 
age. 

2. The gun platform can move at speeds up to 
one-fifth of that of the projectile, and the direction of 
fire may be at any angles of azimuth and elevation 
with respect to the direction of motion of that plat¬ 
form. The bullet, therefore, has an initial velocity of 
departure which is the vector resultant of the velocity 
imparted by the propellant, acting along the bore 
axis, and of the velocity supplied by the moving gun 
mount, acting in the instantaneous direction of mo¬ 
tion. In aeroballistics, then, the initial speed varies 


materially whereas in classical ballistics, a mean 
constant muzzle velocity may be used because mount 
motion is negligible. 

3. The gun-mounting aircraft may operate at any 
altitude from sea level to 40,000 ft. Consequently the 
air density at the point of fire has an important effect 
on the time of flight and the other ballistic quantities. 
In classical ballistics the point of fire is usually at or 
near sea level, and variations in standard conditions 
of pressure and temperature at that point are im¬ 
portant because the ranges are long. 

4. In classical ballistics the axis of a projectile 
makes a small angle with the bore axis which is also 
the direction of departure. The angle between the 
axis and the departure direction is called the initial 
yaw. In fire from airborne guns the initial yaw is fre¬ 
quently large because of the material angle between 
the resultant direction of departure and the bore 
axis of ; the gun. Aerodynamic and gyroscopic ef¬ 
fects 211 are accentuated. The high cross-wind force 
leads to the phenomenon of windage jump via pre¬ 
cession, 142 ’ 143 and the increased drag influences the 
time of flight slightly. 

5. For air-to-air fire, absolute wind — the motion 
of the air mass with respect to the ground — has no 
effect since both the aircraft and the projectile are in 
the same air mass. The effect of relative wind is, how¬ 
ever, most important. Let a bullet be fired sidewise 
from an aircraft and kept in view from the gun posi¬ 
tion. It seems to lag farther and farther behind and 
appears to move in a path that curves rearward. 
Actually, it is moving in a straight line with respect 
to the air mass but fails to keep pace with the firing 
aircraft since it is decelerating because of air re¬ 
sistance. 

1.2 TIME OF FLIGHT 

1.2.1 Reference Systems 

In aeroballistics the bullet is located by one or the 
other of two reference systems. The first system is 
fixed in the air mass with origin at the gun's position 


CONFIDENTIAL 


9 


10 


AEROBALLISTICS 


at the instant of fire. As indicated in Figure 1 the 
position of the bullet at any instant is specified by 
the two Siacci coordinates P and Q. P is measured 
along the line of departure to a point directly above 
the bullet, and Q is measured from this point verti¬ 
cally downward to the bullet. The range covered by 
the bullet in the air mass is approximately P. The 
orientation of P can be given by an azimuth angle A , 



Figure 1 . Aeroballistic reference systems. 


measured in a horizontal plane clockwise from the 
direction of motion, and by an elevation angle E 
measured positively upward in a vertical plane. (It 
is tacitly assumed that the aircraft is flying straight 
and level.) In the second system, the origin (the 
point of reference) moves with the velocity of the gun 
platform at the instant of fire. The bullet’s position 
at any instant is specified by a range D, which is the 
distance between gun and bullet at that time, and 
by the lateral and vertical deviations from the bore 
axis L and Q, called the ballistic deflections. In prac¬ 
tice, D is indistinguishable from its projection on the 
bore axis, and it is this projection (which is the D in 
the figure) that is called future range in ballistic 
tables. In this latter system, the bore axis is specified 
by the azimuth and elevation angles A b and E b . 

1.2.2 Differential Equations of the 
Trajectory 

The basic ballistic equations can be made out most 
elegantly by the methods of vector analysis. 153 In 


Figure 1 the lengths P and Q may be replaced by 
vectors a P and Q, so that if a vector R is introduced 
connecting the origin to the projectile, 

R = P+Q, 


where u is the instantaneous speed of the bullet. 
Assuming that the bullet’s axis coincides with the 
tangent to the trajectory at each instant, i.e., assum¬ 
ing zero yaw, the drag on the bullet caused by air 
resistance is given by 

m — K d u 2 > 

^5 

where m = mass of the bullet, 
p a = air density, 

C 5 = ballistic coefficient of the Type 5 projec- 

tile (c 5 = i — 1 where i depends on the 

shape and d is the diameter), 

K d = drag coefficient (a function of the ratio 
of u to the speed of sound). 

The motion of the bullet is governed, then, by the 
vector equation 

R —- Kdu R + g ) 

c 5 

where g is the acceleration of gravity. The com¬ 
ponents of this equation are the basic equations of 
the trajectory, 

P = — —KduP Q = — — KduQ + g. (1) 

c 5 c 5 

1.2.3 Solution of the Equations 
in the Siacci System 

To integrate these equations, the Siacci approxi¬ 
mation is made as a first step. This approximation 
replaces u by P, which is satisfactory for short times 
of flight. Next, experiments show 144 that K D is given 
very closely by ku~ * for velocities between 1,650 and 
2,950 fps. (The expression for the drag then depends 
on the three-halves power of u.) Using these two 
ideas, equations ( 1 ) may be written 

P = - —k*pi Q = ~ — k*P?Q + g, (2) 

C 5 c 5 


a In the text, a vector is denoted by bold face type and the 
magnitude of the vector by light-faced italic type. In the 
figures, a vector is denoted by underlining and deletion of the 
underline gives the magnitude of the vector. 


CONFIDENTIAL 








TIME OF FLIGHT 


11 


SECONDS 



METERS PER SECOND 


Figure 2. Time-of-flight nomogram. 


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1500 


























































12 


AEROBALLISTICS 


where, now, p is the air density relative to a standard 
ballistic density of 0.07513 lb per cu ft. Equations 
(2) are immediately integrable and yield 





pfc* P 1 
3c>V]^J 


( 3 ) 

(4) 


efficient C which is connected to the quantities in 
equation (3) by the expression 

c _ 0-676p 

c 5 

Typical values of c 5 are given by Table 1. Despite 
the apparent accuracy with which the above values 
of c 5 are stated, it should be appreciated that this 


where P 0 is the velocity of departure. (Notationally, 
Po = Wo.) The value of k* depends on the unit of 
length employed. If P is in thousands of feet, 
= 0.118; if P is in feet per second, k* = 0.00372; 
and if P is in yards, k * = 0.00646. 

The interpretation of the structure of equation (3) 
is interesting. In vacuum p = 0 and as a result the 
initial velocity persists in the P direction. In air, the 
second term on the right of equation (3) is a constant, 
but the reciprocal character of the other two terms 
of the equation means that the constant has little 
effect for small t and P. For a sufficiently large value 
of P the right-hand side becomes zero, but this fact 
cannot be used to establish maximum range since 
the speed would fall below the limit for which the 
three-halves power law is valid. However, this value 
of P determines a vertical asymptote for the hyper¬ 
bolic trajectory asserted by equation (3). In fact it 
is possible to deduce 212 equation (3) by starting with 
the bilinear form 

^ _ a bP 
c -j- dP 

and using the Didion-Bernoulli solution of the bal¬ 
listic problem. 213 The details will not be given. As a 
final remark, in connection with equation (3), note 
that the remaining velocity of the projectile at any 
instant can be obtained 20 by computing dP/dt. 
This velocity is needed in calculations of impact 
energy. 

The derivation of equation (3) depends explicitly 
on the three-halves power law. For projectiles of 
velocity lower than 1,650 fps the more general Siacci 
procedure must be used. 15d For example, a frangible 
projectile has been introduced for training purposes 
which shatters upon impact. 48 To avoid damage its 
velocity must be kept low (below 1,500 fps). For the 
caliber 0.30 Frangible Ball T44 projectile the Siacci 
functions have been computed, 149 so that trajectories 
may be deduced. 

Figure 2 is a nomogram for the computation of the 
time of flight t. It uses the continental ballistic co¬ 


Table 1. Constants for typical ammunition. 


Country 

Type 

Caliber 

Ch 

Muzzle 

velocity 

(fps) 

USA “6 

API M8 

0.50 in. 

0.439 

2,870 

USA 14 <s 

AP M2 

0.50 in. 

0.458 

2,700 

Germany 147 

API, MG151 

20 mm 

0.294 

2,298 

Germany 147 

HE, MG151 

20 mm 

0.180 

2,671 

Germany 147 

HEIT, MG131 

13 mm 

0.243 

2,348 

Germany 147 

APT, MG17 

7.92 mm 

0.345 

2,587 

Japan 19 


20 mm 

0.280 

1,964 

Japan 19 


7.7 mm 

0.234 

2,426 


coefficient varies somewhat from manufacturer to 
manufacturer and even from lot to lot. As it is de¬ 
termined by experimental firings, it also depends on 
the conditions of that firing. 

1.2.4 Time of Flight in the 
Relative System 

The discussion of time of flight may be concluded 
by giving formulas appropriate to the moving co¬ 
ordinate system described in Section 1.2.1. This is 
the natural system to use in airborne gunnery. (The 
Siacci derivation has the technical advantage of re¬ 
ducing the problem to the classical case with a 
properly chosen velocity of departure.) In the rela¬ 
tive system the bore axis may be specified by the 
angle 0 which it makes with the direction of motion. 
Evidently 

cos 0 = cos A b cos E b . (5) 

The speed of departure is given closely by 
Po = V 0 + v G cos 0; 

where v 0 is the muzzle velocity imparted by the pro¬ 
pellant and vg is the true airspeed of the firing air¬ 
craft. The expression 

D = P — v G t cos 0 

is also a good approximation. With the aid of these 
expressions, equation (3) may be written in the form 

PVq cos 6t 2 — ( v 0 — pD)t + D = 0, 


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BALLISTIC DEFLECTIONS 


13 


where 

V = ~~ Vvo ]/ 1 + — cos 0. 

If only terms of the first order are preserved in the 
expansion of the smaller of the two roots of the 
quadratic, the value 

t = — PTi+ pP^n (6) 

Vo-pDl (vo-pDyj w 

is obtained. 19 More refined work would employ two 
terms of the expansion and would correct D by Q sin 
E b for fire at high elevation. 

1.3 BALLISTIC DEFLECTIONS 

1.3.1 Angle Subtended by Gravity 
Drop 

Figure 3 shows the trajectory relative to the mov¬ 
ing gun. The displacement of the trajectory from the 
extended bore axis is called the ballistic deflection. 
This total deflection is decomposed into a lateral de¬ 
flection W measured as a great circle arc in the plane 



Figure 3. Ballistic deflections. 


determined by the bore axis and the direction of 
motion, and into a vertical deflection G measured in 
a vertical plane through the projectile. By Figure 1 it 
is seen that the angle subtended by G at the gun is 


G = 1000 


Q cos E b 
D 


in milliradians. b Substituting Q from equation (4) 


(and investigating numerically the approximations 
for P and P 0 from Section 1.2.4) leads to the value 

- l ./lOOO ll P1C*\ j-y _ 

G= 2 n^-— ) cos ^ (7) 

which is highly accurate for the ranges and speeds of 
tactical importance. 


1.3.2 Trail Angle 


An expression for the angle W subtended at the 
gun by the lateral ballistic deflection may be de¬ 
duced. For beam fire, in the absence of gravity, a 
gun mount in uniform motion is ahead of the bullet 
in time t by the distance 

This is the distance L in Figure 1. The expression for 
this distance arises upon noting that in the absence 
of air resistance the time of flight would be P/P 0 
and the gun mount would remain abeam of the bul¬ 
let. 145 * 1 To obtain the angle subtended by W for non¬ 
beam fire, L must be foreshortened by sin 0, where 0 
is defined by equation (5), and divided by D. These 
operations give 



in milliradians. With the aid of equation (3) and the 
approximation P/P Q = D/v 0 , W can be brought to 
the form 


where 


if =500 ProDdne 
c 5 • E 





pk*DVv o \ 
2 c 5 / 


(8) 


In practice, a constant value is assigned to E by 
supposing that for all-around fire, v 0 = P 0 , and by 
taking an average p of, say, 0.5 and an average D of, 
say, 500 yd. In other words, the main effect of the 
variables p, vg, and D shows in the numerator of 
equation (8). 


1.3.3 Interpretation and Calculation 
of Deflections 


b 1,000 milliradians = 1 radian. In aerial gunnery the ap¬ 
proximations tan a = a and sin a = a are usually made with¬ 
out comment because of the small size of deflection angles. 
(Mils differ slightly from milliradians. 6,400 mils = 360°. 
Hence 1,019 mils = 1,000 milliradians.) 


The trajectory displacement angles W and G will 
always represent corrections to the major aiming 
allowance (which allows for target motion relative 
to the gun mount). In fact, failing any relative target 


CONFIDENTIAL 


















I 0.1 


14 


AEROBALLISTICS 


Ol- 


METERS 


rv, 10 


i i I i i i i i 11 111 i 1111 1 111111111 1 11 h i ii ii I mi lii ml 1111111111111111111111111,1,111,1,1,1,1,1,1,1,1.1,1,1 



METERS PER SECOND 

Figure 4. Nomogram for trajectory. 


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1500 " ^ 25 m 






































THE MOTION IN THE SMALL OF THE PROJECTILE 


15 


motion, they are themselves the aiming allowances or 
leads. For, if a target is pacing a bomber, i.e., main¬ 
taining the same course and speed but not necessarily 
the same altitude, then the target does not move as 
seen from behind the gun. Nevertheless, the bore 
axis must lead the target by the proper values of W 
and G to assure a hit. 

In some cases, such as in experimental studies of 
gunsights, the values of the ballistic corrections must 
frequently be obtained on a mass-production basis. 
It is then desirable to construct special charts, called 
dofographs, 110 for each of which altitude, speed, 
muzzle velocity, and ballistic coefficient are fixed. 
These are specialized tools. More generally, complete 
trajectories can be constructed with the aid of a 
nomogram such as that given in Figure 4, and W 
and G can be calculated by the methods used in the 
above derivations. If such nomograms are carefully 
constructed to a suitable scale, they may well replace 
tables. 148 The advantage lies in the flexibility — C5 
and Vo are not restricted to special values. The dis¬ 
advantage is that auxiliary calculations must be 
made. 

1.4 THE MOTION IN THE SMALL 
OF THE PROJECTILE 

1.4.1 Reasons for the Discussion 

The treatment of the trajectory in Sections 1.2 
and 1.3 was macroscopic in that it described the 
general character of the motion, while neglecting 
certain details. It was assumed that as a projectile 
left the muzzle of an airborne gun, it immediately 
took up the resultant direction dictated by mount 
velocity and propellant velocity and then proceeded 
on a smooth, softly arching curve through the air 
mass. This description and the resulting formulas 
have proved quite adequate for precise investigations. 
Nevertheless, the actual projectile has certain aero¬ 
dynamic and gyroscopic properties which were 
neglected above but which may be briefly exposed. 
The two reasons for such an exposition are: one must 
be certain that the mean trajectory treatment above 
neglects no important effects; and in certain new 
applications such behavior in the small may become 
significant. As an example of the first point, windage 
jump has been calculated and found to be so insig¬ 
nificant for the caliber 0.50 projectile that the effect 
has been dismissed from practical considerations. In 
support of the second point suppose that a large 


caliber gun fires upward at a low velocity — chosen 
low to minimize recoil. 214 The initial yaw is very 
large (45°) and the question of its damping is im¬ 
portant if contact fuses are to function. 

1.4.2 General Features of the 

Motion and Damping of Yaw 

The microscopic motion of a bullet involves three 
effects (1) a vibration of the center of mass with 
respect to a mean trajectory, (2) a precessional mo¬ 
tion of the axis with respect to the center of mass, 
and (3) a drift of the center of mass to the right of 
the vertical plane of departure. All three motions 
occur because of the yaw, which is the angle at any 
instant between the axis of the bullet and the direc¬ 
tion of motion of the center of mass of the bullet. 
More specifically, the first motion is caused by a 
cross-wind force which is analogous to the lift on an 
airplane wing. The second motion is attributed to 
the drag which is-, again, similar to the drag on an 
airplane wing. The third motion is due to gravity 
and the curvature of the trajectory. Yaw, cross-wind 
force, and drag are shown in Figure 5. 



Figure 5. Forces acting on a projectile. (Courtesy 
of John Wiley and Sons.) 


Before explaining the motions described above, the 
nature of the bullet as a potent gyroscope may be 
discussed. This potency is attributable to the high 
rate of spin, which is found by dividing the muzzle 
velocity by the twist (number of feet of barrel for 
one turn of the rifling). (For the caliber 0.50 AP M2 
the spin is 2,160 revolutions per sec.) Over the short 
ranges of aeroballistics it may be supposed that the 
spin rate does not decrease but keeps its muzzle 
value. Figure 5 shows that the bullet resembles a top 
with a point of support at the center of mass and with 
the weight of the top replaced by the drag. By 
analogy with the top, a joint precessional-nutational 
motion of the axis is to be expected. This behavior is 


CONFIDENTIAL 







16 


AEROBALLISTICS 


conveniently pictorialized by giving the pattern 
traced by the tip of the bullet as an epicyclic motion 
associated with two circles (as shown in Figure 6). 



Figure 6. Motion of tip of bullet. (Aberdeen Proving 
Ground diagram.) 


With the notation of that figure, the instantaneous 
yaw is given by 

d 2 = a\ + al + 2 aia ,2 cos ( n\ — n 2 )£, 

where «i is the amplitude of the nutation (of rate wi), 
and a 2 is the amplitude of the precession (of rate n 2 ). 
Both circles decrease in radius during the time of 
flight. This damping of the yaw is equivalent to the 
self-erection of a top. The minimum yaw is a\ — a 2 , 
and, since experiments show that — for calibers up 
to 37 mm at least — if the minimum yaw is initially 
zero it remains zero, it follows that the precessional 
and nutational damping rates are equal, considering 
that the maximum yaw, oq + a 2 , was not zero. The 
damping factor, which multiplies the amplitude, may 
be written in the form 

(1 - S- l )~*e~ pa0tP , 

where S is a stability factor, 143a depending essentially 
on the moments of inertia and the spin of the pro¬ 
jectile, p a is the air density, and a is a constant to be 
determined experimentally. 

1.4.3 Windage Jump 

♦ 

It is clear from the above discussion that the plane 
of the yaw angle rotates. (This is the plane containing 


the axis of the bullet and the tangent to the tra¬ 
jectory.) Since the cross-wind force always acts 
radially outward in this plane it may be expected 
that the center of mass would be pulled around in a 
helical path. It is not so evident that this helical path 
will suffer an overall angular displacement (windage 
jump). The following discussion of the phenomenon is 
heuristic rather than rigorous. 

Point an airborne gun horizontally to starboard 
and fire. Then the initial yaw 6 0 is as indicated in 
Figure 7. Suppose, contrary to fact, 0 that the nuta- 



DIRECTION OF 
PLATFORM MOTION 

Figure 7. Initial yaw. 

tional amplitude is zero, that there is no damping of 
the precessional amplitude, that the effect of gravity 
may be neglected, and that the bullet does not slow 
down. By aerodynamic analogy, the cross-wind force 
may be taken proportional to the yaw, which is 
equivalent to the angle of attack of a wing. Relative 
to the line of departure (Figure 7) measure x hori¬ 
zontally perpendicular to that line and positive to the 
right, and measure y vertically perpendicular to that 
line and positive downward. The coordinates of the 

c The essential physical behavior is not altered by the 
assumptions. 


CONFIDENTIAL 












THE MOTION IN THE SMALL OF THE PROJECTILE 


17 


HORIZONTAL RANGE-M 



HORIZONTAL RANGE* M 



Figure 8. Horizontal projection of trajectory near Figure 9. Vertical projection of trajectory near 

origin. (Aberdeen Proving Ground diagram.) ‘ origin. (Aberdeen Proving Ground diagram.) 


center of mass of the bullet are then (x,y) with respect 
to the departure line. If the precessional period is 
the cross-wind force has as instantaneous com¬ 
ponents in the x and y directions kb cos nt and kb sin 
nt respectively. The equations of motion of the 
center of mass of the bullet are 

d 2 x 

m— = kb cos nt 
dt 2 

d 2 y 

m — = kb sin nt. 
dt 2 

The quantities k , b, and n are constant by assump¬ 
tion. Hence an integration that uses the initial 
values x = y = x = y = 0 gives 

k8 /i * kb / 1 . \ 

X = — - (1 — cos nt) y = — 1 t - sin nt j ■ 

mn 2 mn\ n / 

From the form of x, it is evident that the hori¬ 
zontal projection of the helix d which approximates 
the motion is tangent to the line of departure and lies 
to the right of this line as viewed from the gun. More 
significantly, the form of y shows that the whole 
trajectory has suffered a downward displacement 
through an angle of kb/mnPo radians. This is the 
windage jump. The componential motions are drawn 
after exact calculations in Figures 8 and 9. 

Evident modification of the above analysis for 
port, upward, and downward fire gives the mnemonic: 
if the thumb and first two fingers of the right hand 
are extended at right angles to each other, and if the 
forefinger is pointed in the direction of mount motion 
and the second finger in the direction of fire, then the 
thumb gives the direction of trajectory displacement. 

d The diameter of the helix for small calibers is of the order 
of in. 


It is clear that windage jump is rather a misnomer. 
The phenomenon is due entirely to initial yaw and 
the displacement is not a jump. (In particular the 
random initial yaw of bullets at the muzzle of a 
ground gun due to bore clearance leads to random 
displacement and so builds up a dispersion pattern. 222 ) 

The windage jump, as remarked earlier, is small. 
It may be calculated by the formula 20a 

- b vg . 

J = — b~ sm (9, (9) 

Vo Po 

where b depends on the physical constants of the pro¬ 
jectile. Typically: b/vo is 17 milliradians for caliber 
0.50 AP M2, and is 21 milliradians for caliber 
0.50 AP M8. 

1.4.4 Drift 

To conclude the discussion of the three motions 
listed in Section 1.4.2, drift will be considered here 
only in qualitative terms. The phenomenon is due to 
gravity. Initially the center of gravity starts to move 
downward under gravity while the axis of the bullet 
remains horizontal. The initial yaw attributed only 
to this is therefore in a vertical plane and is small. 
The bullet precesses to the right and down by the 
usual law of gyroscopic precession. Since the tra¬ 
jectory is curved, its tangent slowly rotates down¬ 
ward. For the small yaw under consideration the 
precession can be just slow enough to keep pace with 
the tangent’s downward rotation. That is, the bullet 
keeps pointing to the right of the trajectory. Con¬ 
sequently the cross-wind force is always perpendicu¬ 
lar to a vertical plane and is directed to the right. 
Therefore the bullet drifts slowly to the right under 
this force. (For caliber 0.50 this drift amounts to 
about 7 in. in 1,000 yd and so may be neglected.) 


CONFIDENTIAL 








































18 


AEROB ALLISTICS 


1.5 DISPERSION 

1.5.1 Philosophy of Rapid Fire 

Weapons 

With but few large caliber exceptions, in aero- 
ballistics a burst (a rapid and continuous sequence of 
bullets) is directed at a target. The successive bullets 
will not follow each other in exactly the same path 
and as a result a dispersed pattern is built up at the 
target. The directed burst or pattern principle is 
basic in the philosophy of airborne fire control with 
small caliber ammunition. The expectation is that if 
a large number of bullets are placed rapidly on and 
near a target there will be a high probability that at 
least one will hit the small sub-area vulnerable to the 
caliber being used. That this is really a pollination 
technique is evident from operational statistics indi¬ 
cating that anywhere from 17,000 rounds (B-24, 
Africa, early World War II) to 2,500 rounds (B-29, 
Pacific, late World War II) were expended per 
fighter shot down. In contrast, with large caliber 
projectiles, elaborately and individually directed, 
successive hits can be scored on ships angularly 
smaller than aircraft targets (U. S. Navy, Coral Sea). 

1.5.2 Nature and Statistical De¬ 

scription of the Pattern 

In aerial gunnery only a certain part of the total 
dispersion can properly be attributed to ballistics. 
The situation is somewhat as follows: In tracking and 
firing at an airborne target an instantaneous mean 
point of impact fails to stay on the target since it is 
being carried about by aim wander, deflection errors, 
and instrumental errors. Aeroballistics may discuss 
only this mean point of impact and, even then, suffer 
an enlargement on its concept since a gun mount and 
a gunner will be introduced. This is justified because 
of the influence of these factors on the direction of 
bullet departure. To proceed systematically consider 
a sequence of three ground experiments. 

1. Suppose rounds are fired under precisely the 
same conditions of exact and individual aim. Because 
of initial yaw at the muzzle, manufacturing variations 
in powder charge, bullet shape and weight, and posi¬ 
tion of the charge, al certain pattern will be gen¬ 
erated. But it is a remarkably tight pattern. For 
example, in caliber 0.50 fire through a 45-in. Mann 
barrel a cone of angular diameter 0.25 milliradian 
will contain 75 per cent of the rounds. 


2. If a burst is fired, instead of individual rounds 
as above, with the mount held rigid, elastic vibrations 
of the barrel will occur. The vibration is not simply 
an up-and-down whip. Instead a forced vibration 
occurs and each bullet sensibly leaves in phase with 
the motion of the barrel. Typically, the 75 per cent 
cone is increased to perhaps 2.5 milliradians. 

3. If the mount is not held rigid and if a human 
gunner is asked to hold on a fixed target, the com¬ 
bined mount vibration and gunner error will lead to 
a 75 per cent cone of diameter from 5 to 25 milli¬ 
radians, depending on the installation under test. 154, 

158,161 

It is with this third type of dispersion that one 
normally deals in calculations. In any study, its 
magnitude must be carefully determined for the 
given system even to the extent of noting the effect 
of such variables as aircraft, installation, gunner, 
age of the gun, length of burst. 

When the bullet holes of any burst are collected 
on a flat target normal to the line of fire, they have a 
center of gravity which is called the mean point of 
impact [MPI]. It may be noted that the MPI may 
be systematically displaced from the first bullet due 
to gun jump. Furthermore, with excessively long 
bursts the heating of the barrel may also cause a 
steady shift of the MPI. But, neglecting such effects, 
it is common to consider each round as independent 
of any other so that the dispersion pattern may be 
adequately described by a two-dimensional normal 
or Gaussian distribution with the MPI as mean. In 
practice the variances in two orthogonal directions 
will differ, leading to an elliptic distribution. But 
since the orientation of this elliptical pattern de¬ 
pends in a complicated way on the direction of fire, 
because of different vibrational responses of the 
mounting, it has become customary to replace it by a 
circular distribution (with the two directions sta¬ 
tistically independent). If the MPI is taken as the 
origin, the fraction of a large number of rounds that 
will lie in the cell rdOdr is 

p(r,B) = (10) 

where <j is the standard deviation. 52 Cells of equal 
probability with their centroids are shown in Figure 
10. Since the evidence is that the bullets of a burst 
define a right circular cone, it is usual to take r and 
<7 in milliradians. In military literature the pattern is 
specified by giving the diameter of the circle that 
contains either 50 or 75 per cent of the rounds. 


CONFIDENTIAL 



DISPERSION 


19 


CELLS OF EQUAL PROBABILITY 



.00025 


LINEAR PROBABLE ERROR 


r\j no 


J 11.)1 I 1 . 1 L 1 i l l‘.‘I 


I . i 1 i I 


' 1 1 I ■ ■ 


llilll.ll.ll'.ll 1 I I 


I I I 


LINEAR PROBABILITY 


III 1 


(0 on 


J I I I 


LINEAR AVERAGE DEVIATION 


STANDARD DEVIATION 


L i 


-i-1 L 1 I I 1 1 I 


J i i i I_I—i—i—i—i—1—i_i.i L 


J I. I ..—.l—L 


■ ■ I 


J 1 1.1 i L- 


1-i:l I l.I 


Figuer 10. Cells of equal probability. 


CONFIDENTIAL 


0.4995 





































































































20 


AEROBALLISTICS 


From equation (10) it follows that the diameter of 
the circle containing h per cent of the rounds is 
given by 

d = *1/8 In - 100 . (11) 

' 100 - h v ' 

(For example: if h = 50, d/a = 2.354; and if h = 75, 
d/a = 3.330.) The most elegant specification of the 
pattern size is through a, since a is the radius of that 
annulus of width dr that contains the maximum 
fraction of the total number of rounds composing 
the pattern. 

1.5.3 Change in Pattern Size under 
Trail Gradients and Forward Fire 

Aeroballistic effects can modify the pattern size. 
Suppose that a turret is firing a burst while it is track¬ 
ing at a uniform angular rate. Then the pattern being 
built up at the target, i.e., the pattern relative to the 
gun, is distorted by the change in lateral ballistic 
deflection due to the change in 0. 111 This trail gradient 
is 

dW — 

— = W 90° COS 0 , 
au 

where TF90 0 is the lateral ballistic deflection for beam 
fire, other conditions remaining unchanged. An 
analysis in the vertical direction gives the same 
distortion. It is concluded that 

( , Woo° cos 0 \ 

V 1000 / 

A different type of distortion occurs in forward 
fire from a fighter. 38 A straightforward vector combi¬ 
nation of muzzle velocity and fighter velocity gives 
the expression 

Vo 

G’air — (T • 

Vo T~ V f 

This is illustrated by Figure 11. A change as great 


v r 



Figure 11 . Contraction of pattern in forward fire. 


as 20 per cent can occur, and it should, therefore, be 
taken into account. 

A final point of interest is the aeroballistic effect 
on the dispersion pattern caused by the variation in 


muzzle velocity. This velocity will vary materially 
(100 fps) not only because of manufacturing vari¬ 
ations but also because of the position of the powder 
in the cartridge, the temperature, the age of barrel, 
and the length of burst. 56 Since the aiming allowance 
for target motion is approximately proportional to 
muzzle velocity (see Chapter 2), if all the rounds of 
a burst have a mean muzzle velocity different from 
that assumed by the fire control system, a systematic 
displacement of the MPI from the target will occur. 
But consider now the effect of the variation in muzzle 
velocity around the mean during the burst. The low¬ 
valued bullets will underlead and the high-valued 
ones overlead. The dispersion is increased propor¬ 
tionately along the track of the target, which can be 
regarded as fortunate, since deflection errors such as 
the systematic one described above have their 
greatest value along the track. 

1.5.4 Harmonization 

Because of dispersion it is a reasonable idea, in 
view of the flat trajectories in hand, to attempt to 
remove the necessity of making superelevation allow¬ 
ances for gravity drop in the aiming problem. If the 
guns are set at some fixed slight angle with respect 
to the line of sight, then with the usual condition of 
guns below the gunner’s eyes the trajectory of that 
ideal bullet, which will hit the MPI, will arch up, 
cross the line of sight near the guns, and cross it 
again much later. With a dispersion pattern there is, 
then, an arched cone which supplies a beaten zone 
hugging the line of sight quite closely over ranges of 
tactical importance. In determining the required 
angle of elevation it is necessary to consider: parallax 
(the displacement in the aircraft of the guns from 
the eye of the gunner), gun climb during a burst, 
gravity drop, range to be beaten, and direction of 
fire. 161 Direction of fire is important because nose fire 
to a given future range implies a smaller time than 
does tail fire to that same range. In remote fire con¬ 
trol systems as in the B-29, the physical difficulties 
in carrying out a harmonization scheme determined 
by these considerations are material. 49 

1.6 SUMMARY 

Section 1.1 defines aeroballistics indirectly by con¬ 
trasting certain points connected with fire from air¬ 
borne guns with items in classical exterior ballistics. 


CONFIDENTIAL 











SUMMARY 


21 


Section 1.2 derives formulas for the time of flight 
of a projectile as a function of direction of fire, range, 
altitude, airspeed of the gun mount, ballistic coeffi¬ 
cient, and muzzle velocity. Two coordinate systems 
are used: one fixed in the air mass and the other 
translating with the gun mount. 

Section 1.3 works in the coordinate system relative 
to the gun mount and deduces various expressions for 
the angular displacements of the projectile from the 
bore axis. These displacements are the lateral and 
vertical ballistic deflections. 


Section 1.4 concerns itself with the microscopic 
motion of a bullet with respect to the mean trajectory 
of the previous sections. The phenomena of damping 
of yaw, windage jump, and drift are discussed in 
elementary terms. 

Section 1.5, dealing with dispersion, indicates the 
relation between patterns determined by ground 
firings and patterns arising under air firing. The 
pattern is described by a circular Gaussian distribu¬ 
tion. Harmonization of guns and line of sight is 
mentioned very briefly. 


CONFIDENTIAL 



Chapter 2 


DEFLECTION THEORY 


2.1 INTRODUCTION 

2.1.1 Definition of Deflection 

I n air warfare both the gun platform and the target 
move at high speeds, often at speeds up to one- 
fifth of that of the projectile being used. The high 
speed of the mount, taken on by the bullet, results 
in an appreciable angle between the bore axis and the 
direction of departure of the projectile, and the high 
speed of the target means that it will move a con¬ 
siderable distance during the time of flight of the 
missile. In addition, a bullet has its own ballistic 
deflections. For these three reasons, in order to obtain 
a hit, the bore axis must have, in general, an angular 
displacement from the line connecting gun and target 
at the moment of fire. This angle between bore axis 
and gun-target line is called the deflection or the lead 
(even if it is a lag) or the aiming allowance. The 
theory of aerial gunnery is dependent on a precise 
mathematical account of deflection shooting. Such 
an account is at hand, 15 and it is the purpose of this 
chapter to review compactly this systematic theory. 

2.1.2 Three Problems of Deflection 
Theory 

The problem put to deflection theory is threefold. 
Formulas describing perfect shooting must be de¬ 
rived to serve not only further studies but also to 
supply a norm against which inevitable compromises 
and approximations may be held. Formulas must be 
put in a form amenable to a computation which is 
both rapid and accurate and which has as inputs 
legitimately available data. Finally, the formulas 
must be in a form to indicate suitable mechanization. 
As a combined example of the last two points, legiti¬ 
mate data are those measurable by the gun platform 
at the instant of fire and accordingly available to a 
suggested mechanization (fire control system). 


2.1.3 Conditions for Validity of Theory 

In putting the theory of this chapter to work for 
any situation other than air-to-air fire with present 
ammunition at present day ranges (up to 1,000 yd) 
caution must be observed for three reasons. It is 
assumed that gun platform, target, and projectile are 
all in the same air mass which is in nonaccelerated 
motion. Thus, for a ground target or a ground gun, 
further consideration must be given to the effect of 
the motion of the air mass with respect to the ground. 
In the second instance the detailed consequences of 
various formulas will usually employ a three-halves 
power resistance law for the bullet. Chapter 1 has 
indicated the range of bullet speeds under which this 
law is appropriate. When these conditions are vio¬ 
lated the more exact Siacci treatment of the tra¬ 
jectory must be used. 15g Finally, in most of this 
treatment it is assumed that bullets travel in straight 
lines with respect to the air mass. This means that 
gravity drop, windage jump, and drift are con¬ 
sidered small enough so that they may be accounted 
for terminally by linear superposition. 

2.1.4 The Perfect Bullet 

In general, in aerial gunnery, each bullet of a 
burst requires a personal deflection different from 
that of its fellows because of the continuously chang¬ 
ing conditions of fire. This chapter deals with an 
individual and perfect bullet under general conditions 
of fire. Considerations of dispersion and of proba¬ 
bilities of hits are necessarily deferred. The target 
is a point. 

2.2 NONACCELERATED TARGET 

COPLANAR WITH MOUNT VELOCITY 

2.2.1 Basic Formula 

The plane determined at any instant by the posi¬ 
tion of the target and by the mount velocity is called 


22 


CONFIDENTIAL 


NONACCELERATED TARGET COPLANAR WITH MOUNT VELOCITY 


23 



T IS THE ANGLE OFF OF TARGET AT INSTANT OF FIRE 
Oi IS THE APPROACH ANGLE OF TARGET AT INSTANT OF FIRE 
ARROWS GIVE THE POSITIVE SENSE OF ALL ANGLES 

Figure 1 . Nonaccelerated target coplanar with mount velocity. 


the plane of action. In individual duels between a 
fighter and a bomber this plane of action remains 
sensibly the same during the engagement. (See 
Chapter 3.) For the purposes of this section it is 
necessary to assume only that the target moves with 
constant velocity in the plane of action of the instant 
of fire, since it is clear that a bullet has no relation to 
the motion of the mount subsequent to its departure 
from that mount. The situation contemplated in the 
air mass is drawn in Figure 1 which is suitably 
annotated. With the aid of the dotted line construc¬ 
tions, one readily obtains the formula 

. . vg . v T . 
sin A = — sin t — q — sin a, (1) 

Vo Vo 

where the slowdown factor q is the ratio of the initial 
speed of the bullet, u 0) to the average speed, u, over 
the air range P. 

To justify the name given to q, note that tf = qt 0f 
where tf is the actual time of flight to future position 
and to is the time of flight in vacuum to that posi¬ 
tion. Hence l = q — 1 is the proportion by which the 
time of flight has been increased because of air re¬ 
sistance opposing the bullet’s motion. 

2.2.2 Own-Speed Lead, Fighter 
Lead, and a Classic Theorem 

Two obvious special cases of equation (1) are im¬ 
portant. If either vt or a is zero, 

Vq 

sin A = — sin r. (2) 

Vo 


In this case A is called the own-speed allowance. It is 
significant that the mean velocity of the bullet does 
not appear in equation (2). Only the direction of 
departure is important. Tactically speaking, either 
a target fixed with respect to the air mass or a target 
approaching or receding along the gun-target line 
is under fire. This, then, is the allowance to make 
in strafing a ground target, in the absence of ground 
wind, or the deflection to take against an attacking 
fixed gun fighter, to a first approximation. (See 
Chapter 4.) 

As a second case consider that a first approxima¬ 
tion fighter, whose guns must fire exactly in the 
direction of flight is at the point G. Then the angle 
off, r, must always be equal to the required de¬ 
flection, A. Consequently, equation (1) becomes, 
when the sign situation is thought through, 


v T 

sin A = —q — -sin a , 

Vo + Vg 

or 


sin A --— sin a. (3) 

u 


The minus sign indicates that lead (gun pointing 
ahead of the target) is taken negative. Lag (gun 
pointing behind the target) is given the plus sign 
since most emphasis has been placed on the study of 
fire from a bomber against attacking fighters. 

A comparison of equations (2) and (3) supplies a 
classical, but only approximate, theorem. Place a 
fighter at T in Figure 1, so that there are now two 
gun mounts. For the fighter the approach angle, a, 


CONFIDENTIAL 








24 


DEFLECTION THEORY 


is the same as the angle off, r. And the speed of the 
fighter’s target, v T , is the speed v G of the bomber. 
Finally, the average velocity u of the fighter’s bullet 
is approximately equal to the propellant muzzle 
velocity, v 0 of the bomber’s bullet, since the slow¬ 
down of the fighter’s bullet is compensated for by its 
augmented velocity of departure. Consequently, 
roughly , in a duel between a fighter and a bomber, the 
leads taken are equal in size and opposite in sense. 
The refinements are explored in Chapters 3 and 4, 
and the theorem is put to practical use in Section 
7.2.4. 


2.2.3 Tracking Rate Formulation 

Formula (1) is not adapted to mechanization. It 
requires an estimate of approach angle which cannot 
be made accurately from the gun platform by existing 
devices. It also uses the unknown future range in 
determining the slowdown factor. Finally it is easier 
mechanically to measure the angle between mount 
velocity and the bore axis of the gun rather than the 
angle r. It seems reasonable, in meeting these three 
objections, to work in a reference system translating 
with the mount’s velocity at the instant of fire. In this 
relative reference system the platform velocity is 
given to the target in a reverse sense. When this is 
done, it is evident that the angular rate a> of the gun- 
target line is 

co = — (v G sin t — v T sin a), 

where r is the present range or range at the instant of 
fire. With this formula a can be removed from 
equation (1), thus meeting the first objection above. 
The result is 


no v G . 

sin A = q — — l — sm r, 

Vo Vo 


where l = q — 1. To meet the third objection, take 
7 = t + A, where y is the angle between mount 
velocity and the bore axis. Then 


where 


and 


sin A = t m cc — b sin y, 
qr 


tm — 


Vo — lv G cos y 


(4) 

( 5 ) 


of t m appears in Section 5.3.5.) A prefatory discussion 
of the important formula (4) is, however, in order 
at this place. 

2.2.4 Kinematic and Ballistic 
Decomposition 

To proceed obliquely with this discussion of 
formula (4), the analogue of Figure 1 is shown in 
Figure 2 which uses the reference system relative to 
the gun mount. In that figure, v R is the relative target 



Tf 


r IS THE PRESENT RANGE 
D IS THE FUTURE RANGE 

Figure 2. Deflections relative to the gun mount. 

speed, which is the vector sum of v T and — v G , and i p 
is the approach angle of the relative path. In the 
absence of lateral ballistic deflection (trail), the gun 
points at Tf, and the corresponding aiming allowance 
A* is called the kinematic deflection. But the tra¬ 
jectory curves rearward when viewed from behind 
the gun. Hence the ballistic correction A 6 must be 
made, gravity being neglected at this point. In this 
system, the total deflection is expressed, then, in 
the form 145 

A = A* — Aft. (7) 

It will be appreciated that this deflection must be 
the same as the deflection given by equation (4). A 
direct identification of equation (7) with equation 
(4) can be made. From Figure 2 it follows that 

• * v ^s • , 

sm A k = sin \p • 

But ru = v R sin ^ and D/t f = v. Hence 
rco 

sm A k = — , 
v 

where v is the average speed of the bullet over the 
future range D. Using Figure 3, in which auxiliary 
lengths x and y have been introduced, it is easy to 
show that 


, lv G cos A 

& — ~ • ( 6 ) 

Vo — lv G COS 7 anc J 

The second objection, concerning the unknown future 
range, must be dealt with carefully. (The treatment 


q 2 v 2 = (v 0 — lv G cos 7) 2 -f- Pv G sin 2 7, 


sin At = 


lv G sin 7 
qv 


CONFIDENTIAL 









ACCELERATED TARGET COPLANAR WITH MOUNT VELOCITY 


25 


But ( Ivg sin y) 2 is of negligible size compared with 
(v 0 — Ivg cos t) 2 . Hence qv may be replaced by 
{vq — lv G cos y) ; and simple substitution now identi¬ 
fies equations (7) and (4) by 

A * = t m u, A b = b sin y, 

provided sin A k and sin A b are replaced by A k and A* 
measured in radians, and cos A is taken to be unity 
in equation (6). 

2.2.5 Time-of-Flight Multiplier 

The quantity t m of equation (5) should be examined 
closely. It has the dimensions of time but since it is 
present range divided by mean velocity over future 
range it has no direct physical meaning. It is called 
the time-of-flight multiplier. We may write 

. _ r _ r v p v p 

lm ~ _ — ~ _ — T - L P’ 

V V p V V 

where v p is mean velocity over present range and t p is 
the time of flight over present range. If the relative 
motion path is incoming, i.e., D < r, it follows that 
v p /v < 1 and tpu is slightly too large a kinematic de¬ 
flection. The reverse is true if the relative motion 
path is receding. 251 In a mechanization designed for 
all-around use, the ballistic computer would simply 
translate present range into present time of flight. 
But if the tactical circumstances were such that 
targets were always closing, a correction factor should 
be applied. This is a first and simple example of a 
basic principle in fire control theory. The time-of- 
flight multiplier need not fit any particular ballistic 
table but should be chosen as a function of range in 
such a way as to optimize performance over an ex¬ 
pected set of tactical situations. 41 

2.3 ACCELERATED TARGET COPLANAR 
WITH MOUNT VELOCITY 

2.3.1 Derivation of Acceleration 
Correction 

For the tactically important case of an accelerated 
target, which means a curved path, a change in 
speed, or both, the formulas of the previous section 
must be modified by the addition of a correction 
term. If the chord of the target’s path segment from 
T to T f makes an angle a with the gun-target line, 
and if the average speed of the target over this 


segment is vt, then equation (1) may apply and is 
written a 

v G . Vt . - 

sin A = — sin r — q — sin a, 

Vo Vo 

where Vt = TT f /tf. Conceptually, the actual target 
has been replaced by an equivalent one, as far as 
impact is concerned, whose approach angle is the 
average approach angle of the actual target. An 
estimate of v T is given by (see Section 2.4.2) 

Vt = Vt ~ f- \tf * Vt, 

where vt and v T are evaluated at the moment of fire. 
If one assumes that the change in path curvature 
is not radical, over the times of flight contemplated 



AB 1 V 0 GT f = c 

605 «o BC = X 

GA * V G CTf = y 

AT f * v f 

Figure 3. The firing parallelogram. 

— and this is a valid assumption — the target may 
be taken to move in a circle at constant speed Vt and 
to turn through an angle 2(a — a) during t f . If the 
angular rate of change of the target is cor, the angle 
through which it turns is also cor • t f and so 
a = a Jcor • tf. 

Consequently the deflection formula becomes 

. . Vg . v T + \vt • tf . 

sin A = — sin t — q -sin (a + §c o T • tf) , 

Vo Vo 

or 

. Vg . Vt 

sin A = — sin t q sin ol \qtfp , (8a) 

Vo Vo 

where 

Vt 

p = — sin (a + §cor • t f ) 

Vo 

^ 2 vt I ” 

vo L 

and p is approximated by 

vt Vt 

p = — sin a + — cor cos a. (8b) 

Vo Vo 


a To follow this discussion by figure, modify Figure 1 by 
drawing an arc through T and T f and placing bars over a 
and v T . 


sin (a + ^cor • tf) — sin a "I 

~7 f } 


CONFIDENTIAL 










26 


DEFLECTION THEORY 


If it is assumed that the gun mount is moving at 
uniform velocity, then vg = 0. Since r = o>, it follows 
from the physically evident rate relations 


that 


ro) = Vq sin r — Vt sin a 
—r = Vq cos t + Vt cos a 
a = cot — u, 



( 9 ) 


( 10 ) 


where M is the angular momentum r 2 co of a target of 
unit mass. [Formula (10) may be verified by calcu¬ 
lating the derivative of r 2 o>.] With the aid of formula 
(10) and the first equation of (9), equation (7) can 
be written 


where 


, Ora 7 v a . 

sm A = h - l — sin r 

Vq Vq 


h = 


1 + 


t f • M 
2 M 


(id 


Finally, as in the derivation of equation (4), one may 
take 

sin A = ht m oi — b sin y (12) 


where t m , the time-of-flight multiplier, and b, the 
ballistic deflection factor for beam fire, are exactly 
as in equations (5) and (6). Conceptually, equation 
(12) is quite important. It shows that deflection 
formulas for an accelerated target can be obtained 
from those for a target in uniform motion by multi¬ 
plying the time-of-flight multiplier by a (non¬ 
constant) factor h. In the design of eye-shooting 
systems and own-speed sights, and in the calibration 
of lead computing sights this is decisive. 


Put, as above, M = r 2 oo and expand M in a Taylor’s 
series about t = 0. Then 


A = — f (M -|- Mt -f- • • ‘)dt 
2 Jo 

— i(Mtf + %Mtf + • • •), 



FUTURE TARGET POSITION T f 
RELATIVE TO GUN 


Figure 4. Accelerated target, relative to gun mount. 


where, now, M, M, • • • are evaluated at t = 0. Since 
tf is small, terms of the third and higher orders may 
be neglected. An independent evaluation of the area 
A is given by the area of the triangle GTT f , since t f 
is small and the path curvature cannot be large for 
aerodynamic reasons. The area of the triangle is 
A = \rD sin A*. 

When this is equated to the previous expression, one 
finds that 

sin A* = ht m co, (13) 

where the letters have the same meaning as in the 
earlier discussion. Finally, since the argument based 
on Figure 3 uses only the impact point and is not 
concerned with the target’s meanderings in reaching 
this point, it applies immediately here. The identi¬ 
fication of the air mass coordinate formula and the 
kinematic-ballistic decomposition is complete. 


2.4 THE GENERAL THEORY OF 
DEFLECTION 


2.3.2 Kinematic and Ballistic 

Decomposition 

To parallel the discussion of Section 2.2.4, the situ¬ 
ation here may be considered in the reference system 
relative to the mount. This treatment emphasizes the 
role played by the angular momentum of the target 
on its relative path. From Figure 4, it is evident that 
the area A swept out by the gun-target line during 
the time of flight of the bullet is 

1 r l f l r l f 

A = — I r 2 rdt = — I r 2 wdt. 

2 Jo 2 Jo 


2.4.1 Extended Conditions 

A general treatment must permit the gun mount 
and the target to move in arbitrary space paths. It 
must also allow the bullet to move in a vertical plane 
rather than in a straight line. For the first require¬ 
ment it may be said that the paths are smooth in the 
analytic sense and curve gradually in the tactical 
sense. By implication, one will not be concerned with 
derivatives of the third and higher order. Under the 
second requirement gravity drop is permitted. Al¬ 
though deviations normal to the vertical plane that 
contains the direction of departure of the bullet are 
not permitted here, the methods of the section may 


CONFIDENTIAL 





THE GENERAL THEORY OF DEFLECTION 


27 


be generalized should missiles of the future require 
consideration of such behavior. For the three-dimen¬ 
sional situation, the evident tools are vector algebra 
and vector calculus. 1515,216 The methods are par¬ 
ticularly applicable to support fire situations and to 
aerodynamic lead pursuit curves (Chapter 3). Finally, 
since the methods are vectorial, specialization to 
various coordinate systems corresponding to par¬ 
ticular decompositions of deflection is readily accom¬ 
plished. 

2.4.2 Derivation of General Formula 
by Vector Methods 

In the air mass the trajectory is described by a 
vector b R connected to the Siacci vectors P and Q 
(see Section 1.2.2) by 

R = P + Q. 

Let r be the gun-target vector, let v G and vt be the 
velocities of mount and target, and let v 0 be the 
propellant muzzle velocity (in the direction of the 
bore axis). All vectors are functions of time measured 
from the instant of fire. In the air mass reference 
system, one writes triangularly 

r+f t 'v T dt = R- (14) 

Jo 

Holding this relation in reserve for the moment, 
it is trivial to observe that the bore axis is placed in 
lead position by rotation from the gun-target line 
through the appropriate lead angle. But this rota¬ 
tion suggests the representation of deflections by 
vectors perpendicular to the bore axis and to the 
gun-target line and hence implies cross products. By 
definition 


x r x v 0 

Ao — 

Ao = sin A 

rv o 

N Vg x r 

> v g • 

X<? = 

\ G = — sin r 

rv o 

Vo 

% v T x r 

. Vt . 

A T — 

At = — sin a. 

rv o 

Vo 


Now u 0 = v 0 + v G , and u 0 = UoP/P. Hence from 
equation (14) 

r 1 Q 

~~ ~\r vt = — (vo + v G ) + — f 

_ t/ q t f 

b The dot product of vector A by B is the scalar A - B = AB 
cos 9 where 9 is the angle between A and B. The cross product 
of A by B is written A x B and is a vector perpendicular 
to the plane of A and B, with a positive sense according to 
a right hand system (A, B, Ax B). The magnitude of 
A x B is AB sin 9. 


where q = Uo/u, u = P/t f , and t f v T = 
multiplication by q there arises 

Uq 

Vo = —Vo + qv T - W + - r, (16) 

where w = u 0 Q/ P. If the cross product of equation 
(16) by r/r on the left is taken it follows that 

Ao = X(? — q^r + X<7, (17) 

where the list in equation (15) is now extended by 



- vr x r 

A T — 

rv o 


- v T . - 
At = -Sin a 

Vo 


o> X r 

As = — 
rv o 


X g 


CO . 

— sin Z, 

Vo 


(18) 


and where a is the average approach angle over tf 
and Z is the angle between r and the true vertical. 
Formula (17) makes precise one’s intuition about 
the form of the total lead and clearly reduces to 
equation (1) for the two-dimensional case without 
gravity. 

To obtain the appropriate generalization of equa¬ 
tion (11), consider v T in detail. A good approxi¬ 
mation to vt is given by the expression 

Vt = Vt + \tfVTy 

where v T and v T are evaluated at t = 0. This arises 
by taking three terms in the series expansion of the 
vector giving target position with respect to origin 
of the air mass coordinates, differentiating, and 
integrating from 0 to tf. 

Turn now to the reference system relative to the 
gun. The velocity of the target in this system is 
v R = v T — v G . The angular momentum of unit mass 
at the target position is 

M = rxr = rx v R , 

and its derivative is given by 

M = rx r = r x v R . 


The angular velocity co of the gun-target line is con¬ 
nected to M by 

M = r 2 o>. 

Using these relations one finds that 

\t x r = — M + rvo\ G — + rvo\tf\ Gl 

where 

xX = 


CONFIDENTIAL 










28 


DEFLECTION THEORY 


Finally, substitution in equation (17) yields the vec¬ 
tor analogue of equation (11) 

. f M M1 qrco 

“ |_ m + 2 ^/m J ~ ~ (19) 

(The term involving X^ means that accelerated 
mount motion is permitted.) All quantities in equa¬ 
tion (19), except t f , are evaluated at the instant of 
fire. Furthermore, the effects of acceleration are 
explicitly exhibited in amount and direction. Equa¬ 
tion (19) can be regarded as an expression of suitable 
generality for all fire control applications contem¬ 
plated in this account. The technical ease with which 
it is obtained is noteworthy. 

2.4.3 Special Coordinate Systems 

In the applications, formulas such as (19) must be 
expressed in particular coordinate systems. The four 
most common systems are 15c (1) azimuth and eleva¬ 
tion, (2) sight elevation and traverse, (3) gun ele¬ 
vation and traverse, and (4) parallel and perpendicu¬ 
lar to the plane of action. 

In the first system a fuselage axis and a wing-span 
axis determine an azimuth plane. The elevation plane 
is perpendicular to the azimuth plane. The gear 
system of the usual turret not only illustrates this 
system but shows that the azimuth plane does not 
necessarily contain the aircraft’s velocity vector (the 
aircraft may be flying nose high or nose low) and that 
zenith in the system is not necessarily the zenith 
with respect to the earth (the aircraft may be diving 
and turning). If I is a unit vector in the forward 
direction of the fuselage axis, if K is a unit vector 
directed outward along the starboard wing, if J is a 
unit vector directed upward with respect to the air¬ 
craft, and if ri and v 0 i are unit vectors in the direc¬ 
tions of gun-target line and bore axis respectively, 
then 

Ti = (I cos A K sin A) cos E + J sin E 
Vox = (I cos A 0 + K sin A 0 ) cos E 0 + J sin E 0 , 

where A and A 0 are azimuths in the system, and E 
and E 0 are elevations (of gun-target line and bore 
axis respectively). The azimuth lead and elevation 
lead are given by 

A a = A 0 — A , A e = Eq — E. 

For the second (sight) system, let tt S e be the plane 
containing r and perpendicular to the azimuth plane, 
and let tst be the plane containing v 0 and per¬ 
pendicular to ttse • If c is a unit vector along the 


intersection of these two planes, then the angle be¬ 
tween ri and c is called the sight elevation lead, A se, 
and the angle between c and v 0 i is called the sight 
traverse lead, A ST . One has the formulas 

sin A st = sin A a cos E 0 , 

sin Eq = cos A S t sin (E + A se). 

The third (gun) system is similar to the second in 
that 7 tge is a plane containing v 0 and perpendicular to 
the azimuth plane, and ttgt contains r and is per¬ 
pendicular to tge- Then 

sin A gt = sin cos E, 

sin E = cos Agt sin (E 0 — Age). 

The fourth (plane of action) system uses a plane 
t r containing v G and r. If 7 r 0 is a plane containing v 0 
and perpendicular to 7 r, and if c is a unit vector de¬ 
termined by the intersection of these two planes, 
then the angle A y between ri and c is called the lead 
in the plane of action and the angle A_l from c to v 0 i 
is called the lead out of the plane of action. This 
system is particularly useful in theoretical studies 
since many of the tactically significant situations 
occur in a sensibly fixed plane of action and so A y 
accounts for most of the total deflection. The other 
systems are appropriate for mechanizations of the 
fire control problem. 

The discussion in this section will be restricted to 
the gun elevation and traverse system, which is 
typical of the last three systems in that deflection is 
obtained by a pair of rotations in orthogonal planes. 
Introduce unit normals c T and c E to ttgt and 7 tge 
respectively, so directed that 

c x Ti = Ct sin A gt, c x v 01 = c E sin Age • 

Then the leads Agt and Age are computed by the 
formulas 

sin Agt cos A G e = c T • (ri x v 01 ) = c T • X 0 

sin Age = c E • (ri x v 0 i) = c E • X 0 . (20) 

When equation (20) is combined with formulas such 
as (19) suitable expressions for A G t and A G e arise 
after rather massive manipulation . 150 

2.4.4 Gun-Roll Error 

Sights such as the K-3 use gun tracking rates and 
base their mirror system on the gun. Hence the gun 
elevation and traverse system is appropriate in de¬ 
scribing such equipment. But this system has a 
peculiarity, accentuated with elevation, which leads 
to an appreciable error in deflection. 21b The angular 


CONFIDENTIAL 




SUMMARY 


29 


velocity w of r has components w r and o) E corre¬ 
sponding to rotations of the gun-target line in the 
traverse and elevation planes. Now consider the gun 
coordinate system. It will have angular velocities 
wo , w 0 t, and oj oe . But in systems measuring gun 
rates, only w 0 r and w 0 e are obtained. The roll of the 
bore axis, w 0 , is not measured. Hence the angular 
speeds assigned to r by the gun rates are 

C Ct — COo T — A GT, We = CdoE — A GE, 

since the rate of change of deflection gives the relative 
speed of gun-target line and bore axis. But co 0 should 
also contribute to co r and a E . The resulting errors are 
called the gun-roll errors and are given approxi¬ 
mately by 15f 

€gt = — Aer Age tan Eq, €ge = A 2 gt tan Eq. 

If A gt is 0.1 radian, then for elevations greater than 
45°, the elevation gun-roll error is greater than 10 
milliradians, which is significant. 

2.4.5 Timeback Method 

This chapter devoted itself to the computation of 
deflections based as much as possible on present 
data, since these quantities are all that are available 
to a fire control system at the instant of fire. In ex¬ 
perimental and theoretical studies it usually happens 
that complete knowledge of the paths of gun mount 
and target is available. In such cases one may choose 
to circumvent the above formulas and return to first 
principles. That is, one can start by choosing a point 
of impact. Then either future range or air range is 
known exactly and therefore time of flight and kine¬ 
matic deflection are known, as is the required posi¬ 


tion of the bore axis to generate the assigned point 
of impact. Hence the present position corresponding 
to the chosen point of impact is determined since the 
path is expressed as a function of time. A table can 
be built up of deflection versus the time parameter 
of the path in which one may interpolate at pleasure. 
The method is usually called the timeback method. 
It will be discussed again in Section 7.3.3. Whether 
formulas or timeback are used in the calculation of 
deflection, systematic computing aids in the form of 
tables and graphs are available. 15a 


2.5 SUMMARY 

The introduction, Section 2.1, gives the following 
reasons why a systematic theory of deflection is re¬ 
quired (1) to supply norms for approximations, 
(2) to furnish rapid computational procedures using 
legitimately available data, and (3) to determine 
the nature of possible mechanizations. 

Section 2.2 considers the simple case of a target, 
moving at constant speed, on a straight line which 
lies in a common plane with the gun-mount velocity. 
The formulas deduced will be used repeatedly in the 
remainder of the text to elucidate various points. 

Section 2.3 permits the target to move in a curved 
path. The form of the acceleration correction factor 
is computed. This section is a foundation for Chap¬ 
ter 4. 

Section 2.4 uses vector methods to treat the general 
problem in which mount and target may move in 
arbitrary space paths. Particular coordinate systems 
used in fire control are discussed. A brief explanation 
of the phenomenon of gun roll is given. 


CONFIDENTIAL 



Chapter 3 


PURSUIT CURVES 


3.1 INTRODUCTION 

3.1.1 Pursuit Curves in Modern 

Warfare 

T hebe are many situations in which an object 
moving along a path of its own choosing is pur¬ 
sued by another object moving in a path constrained 
to point instantaneously either directly at the pur¬ 
sued (pure pursuit), or at some point in the vicinity 
of the pursued in accordance with some definite law 
(deviated pursuit). In one homely and classical ex¬ 
ample, the pursuer is a dog in a field and the pur¬ 
sued is the dog’s master who walks along the edge 
of the field. If the dog were blind, he might run 
toward the sound of a whistle blown continuously by 
his master. The former case is pure pursuit, and the 
latter is an example of deviated pursuit. 

In modern warfare pursuit curves arise in three 
types of situations: 

1. Homing missiles may continuously change 
heading under radio, optical, or acoustic guidance 
unwillingly supplied by the target. 

2. Aircraft, directing rockets or large-caliber pro¬ 
jectiles at fixed ground targets, may find themselves 
in an air mass moving with respect to the ground. 
If the motion of the air mass is reversed and then 
given to the target a pursuit curve arises. 

3. The standard fighter aircraft of World War II 
employed a heavy battery of guns fixed in the air¬ 
craft to fire sensibly in the direction of flights To 
change the direction of fire the aircraft itself must be 
flown in the new direction. Consequently, unless the 
fighter is directly behind or ahead of its airborne 
target, it must, ideally, fly on such a correctly banked 
turn that the correct and changing aiming allowance 


a Hence, fire from fighters is called fixed gunnery. If guns 
can be positioned freely with respect to the direction of flight, 
flexible (or free) gunnery arises. This is usually true for 
bombers. 


is continuously made. b By doing this the bullet pat¬ 
tern is held on the target until destruction is ef¬ 
fected. 194 

Case 1 has been studied with the purpose, among 
others, of determining turning rates, since certain 
missiles have control limitations in this respect. 
Case 2 has been considered in assessing the effect of 
the path on the aiming problem (rockets), and in 
determining terminal conditions for bomb release 
(75-mm cannon-firing path for the B-25H). These 
two cases do not come within the scope of this ac¬ 
count of aerial gunnery. An appreciable number of 
the techniques exposed below are, however, appli¬ 
cable to such studies. 

3.1.2 Reasons for an Elaborate 
Investigation 

The rationale of the elaborate pursuit curve in¬ 
vestigations outlined in this chapter is discussed 
below. 

1. The primary function of the defensive gunnery 
of a bomber is to prevent its parent bomber from 
being shot down by attacking fighters. Consequently 
it is reasonable to require the greatest accuracy of 
the defensive fire control under the circumstances of 
the ideal attack described as Case 3 above. The fire 
control must predict the future position of the target 
on such courses quite closely. To determine future 
positions requires the computation of aerodynamic 
lead pursuit curves. [It will be made clear in the 
sequel how such computations affect (1) the gener¬ 
ation of position firing rules of thumb (Chapter 4), 
(2) the choice of percentage for an own-speed sight 

b The alternative for the attacker is to hold a fixed direction 
of flight so arranged that the target will fly through a stream 
of bullets. Because of the rapidity with which the target will 
pass through the bullet stream and the spacing between suc¬ 
cessive bullets, very few hits may be scored. This strafing 
attack has been dismissed by all air forces as tactically in¬ 
efficient in view of the limited vulnerability of the target to 
present calibers and rates of fire. 


30 


CONFIDENTIAL 




THE ELEMENTS OF PURE AND DEVIATED PURSUIT THEORY 


31 


in a particular tactical circumstance (Chapter 4), 
and (3) the calibration of the time-of-flight setting for 
rate type sights (Chapter 5).] 

2. From the point of view of offensive or fixed-gun 
fire control a study of pursuit curves leads to an 
appreciation of the effect of aerodynamics on the 
aiming problem, and to appropriate calibration of 
the lead computing sights used by fighters. 

3. Pursuit curve studies have culminated in the 
detailed analysis of Section 3.4. Frequently one 
wishes to use the simpler methods of Sections 3.2 
and 3.3. Since the complete picture is at hand, 
approximations may be tested as desired. 

4. Finally, the design in the large of defensive fire 
control systems requires a knowledge of those limita¬ 
tions and possibilities of fighter attacks such as are 
given in Section 3.5. Design in the large means, here, 
the choice and disposition of armament and the op¬ 
timization of performance of the fire control system 
over the appropriate range-azimuth-elevation cells. 

3.2 THE ELEMENTS OF PURE AND 
DEVIATED PURSUIT THEORY 

3.2.1 Assumptions and Coordinates 

It will be assumed throughout this section that the 
pursued chooses to follow a straight line path at a 
constant speed Vb . Further, neglecting aerodynamic 
effects or deliberate throttle variation, the speed of 
the pursuer is also to be a constant v F . (The velocity 
subscripts indicate that a bomber and a fighter are 
to be the objects of the primary physical realization.) 
If the laws of deviation are restricted to those in 
which the pursuer always homes on some variable 
point on the pursued’s track, it follows that the 
curves considered must be plane curves. The plane 
containing the two tracks is called the plane of action. 
Under the assumptions made above, the geometrical 
results are independent of the angle between the 
plane of action and a horizontal plane through the 
pursued’s track. 

In studying these curves, the most natural co¬ 
ordinate system to use is an itinerant one whose 
origin is held on the moving object under pursuit. 
In such a relative coordinate system, the velocity 
vector of the pursued is transported to the pursuer 
and there reversed in sense. The pursuer then moves 
as dictated by the vector resultant of its own velocity 
and this reversed velocity. 

Appropriate tactical variables are the range p and 
the angle 0 measured positively from the rearward 


track of the pursued up to the line joining the par¬ 
ticipants. Experience has shown that Cartesian co¬ 
ordinates are not efficient. 

3.2.2 Equations for Pursuit Curves 

The above circumstances are put down in Figure 1. 
The deviation angle 5 is specified separately as some 



Figure 1 . A typical instant of deviated pursuit. 


function of 0. (More generally, it would be a function 
of both p and 0.) Three instances of such functions 
concern us in this section. They are (1) 5 = 0, which 
is pure pursuit, 8 e.g., homing by a connection through 
light or radio; (2) 5 = constant, which is fixed lead 
pursuit, e.g., a fighter using a fixed average lead in 
attacking a bomber c ; and (3) 5 = v sin 0, which is 
a variable lead pursuit, 8a e.g., when v is a positive 
constant, a fighter attacks with a variable lead, com¬ 
puted on the assumption that his ammunition has 
a mean fixed velocity over the ranges in question; 
and, when v is a negative constant, a missile homes 
acoustically on a target, 180 so that v is the constant 
ratio of the speed of the pursued to the speed of 
sound. Later sections successively elaborate the 
deviation function to make it more realistic. 

For these three cases the equations of the relative 
path may now be deduced. From Figure 1 the rate 
of range closure is 

dp 

— = — v F cos 5 + v B cos 0, (1) 


c Certain Japanese documents 188 indicate that this was 
doctrinal procedure for the Japanese Air Force during World 
War II. 


CONFIDENTIAL 








32 


PURSUIT CURVES 


and the rate of rotation of the range line about the 
pursued as pivot is 


dO 1 . 

— = — (v F sin 5 — v B sin 0). 
dt p 


( 2 ) 


The effect of the pursued’s speed, as shown by the 
last equation, is to cause the pursuer to crab toward 
a point astern of the pursued. The pursuer is said, 
picturesquely, to be “sucked flat.” When equation 
(1) is divided by equation (2), time vanishes and the 
speed ratio p = vf/v b appears as a natural param¬ 
eter. The equation 


dp —p cos 5 + cos 
P 


p sin 8 — sin 


dO 


( 3 ) 


is integrable for the deviation functions in question. 
The results are: 


Pure pursuit 


tan M 0/2 
sin 0 


Fixed lead pursuit 


1 + 


1 — p sin 5 0 
Lsin 0 — p sin 5 0 J 

| A + MSi 

' 1 — p si 


( 4 ) 


( 5 ) 


sin 5 C 
sin 5 C 


(t-t) 


l-l/ l ± M sin • tan /±_ 

V 1 - m sin d 0 V 2 4 / 

Variable lead pursuit 221b 

p ["/cos 5 + v cos o\* v/2 

Po LVcos 5 — v cos 0/ 

'cos 5 — cos 0 


n cos 5o 


\/1 —^2 sin 2 do 


/cos 5 — cos tfV /2 i 1 1 

\cos 5 + cos 0/ sin p J 


1/(1 — Mv) 


( 6 ) 


For the variable lead pursuit a close and useful 
approximation is obtained by putting cos 5 = 1 in 
equation (3). The simpler formula is 


ftan M 0/2 “I i/(i -m*0 


L sin 0 J 


( 7 ) 


It is clear that equation (5) collapses into equation 
(4) when 5 0 = 0 and that equations (6) and (7) also 
reduce to equation (4) when v = 0, i.e., when the 
bullets used have infinite velocity. When 0 = 90°, 
p = Po in each of these four equations. Consequently 
Po is another natural parameter — the 'proximity 


parameter. It is the range on the beam, and each pur¬ 
suit curve can be extended backward or forward as 
required to give its characteristic p 0 . 

Figure 2 illustrates the relative positions of these 
three types of curves, and Figures 3 and 4 supply 
convenient nomograms for the computation of pure 
pursuit courses. A large number of pursuit curves 
have been computed and graphed. 8 - 141> 180> 221 

Vp * 450 FPS v 0 = 300 FPS 
P 0 = 3300 FT 

90° 



3.2.3 Bifurcated Pursuit 

In the preceding derivation the factor v was the 
ratio of the pursued’s speed to the speed of the bullet 
fired by the attacking fighter [the deviation function 
comes, in fact, from eq. (3), 2.2.2]. Normally, there¬ 
fore, v will be much less than 1. Certain implications 
of v > 1 are worth exploring with conceivable situa¬ 
tions of the future in mind. Attack must be from a 
forward direction if capture is to result, and for this 
region two separate pursuit curves are quite possi¬ 
ble. 215 This is demonstrated most easily by the con¬ 
struction of Figure 5. In this figure v = v B /v and is 
greater than 1. After drawing the boundary line, a 
semicircle of any convenient radius R is constructed 
as shown. The line connecting the pursuer F to the 
pursued B cuts the circle at Pi and P 3 . The lines PP 2 


CONFIDENTIAL 

























THE ELEMENTS OF PURE AND DEVIATED PURSUIT THEORY 


33 




\ 


1 , 20 - 


1 , 15 - 


t 


VO- 


1 , 05 - 


1 , 00-3 



Figure 3. Nomogram for pure pursuit. 


CONFIDENTIAL 







34 


PURSUIT CURVES 



CONFIDENTIAL 







THE ELEMENTS OF PURE AND DEVIATED PURSUIT THEORY 


35 


and FP a are parallel, respectively, to P\M and P$M. 
The pursuer can move in either of these two direc¬ 
tions. Consider PP 2 . Then we should have 

v sin 6 FP 2 
v b sin 8 BP 2 

But the construction achieves this since 
FP 2 _ PiM _ R_ _ . _v_ 

BP 2 = BM = BM~ Sm<t> ~ Vb' 

And the alternative has exactly the same treatment. 



Figure 5. Bifurcation of pursuit. 


3.2.4 Methods of Introducing 
Time as a Parameter 


The Local Method 8b 

Both p and 6 may be expanded in power series 
valid in the neighborhood of any specified point on 
the curve. The derivatives required in these expan¬ 
sions are readily obtained by repeated differenti¬ 
ation of equations (1) and (2). Over intervals of time 
corresponding to normal times of flight of projectiles, 
the convergence is rapid. 

The Midpoint Method 

This procedure renounces analysis and reverts to 
an approximation to the geometrical definition of the 
curve in question. It will suffice to sketch the method 
for a pure pursuit curve. Working in a fixed coordi¬ 
nate system, determine the position of the pursued at 
intervals of, say, one-quarter second. Over the first 
time interval let the pursuer move in a straight line 
from his initial position toward the midpoint of the 
first interval of pursued’s motion. This yields a new 
position for the pursuer from which he can move in a 
second straight-line segment over the next time 
interval toward the midpoint of the second interval 
of pursued’s motion. Continuing in this fashion a 
table of positional values given explicitly in terms of 
time is built by the most elementary of computing 
means. (This method has been used extensively in 
the production of synthetic motion pictures for use 
in flexible gunnery training devices. 201 ) 


A basic disadvantage of the solutions (4) through 
(7) of the pursuit curve problem is that range and 
angle off are not given explicitly as functions of time. 
It is frequently necessary to have such dependence 
at hand. For example, in determining exactly the 
deflection to take against a fighter on such a curve 
one must, by cut and try, match up time along the 
curve from a chosen present position to the required 
future position of impact, with the time of flight over 
the range to this future position. There are three ways 
of getting points on the curve labeled with the 
appropriate time: 


The Implicit Method 


If p in terms of 6 is substituted in equation (2), 
upon integration t as a function of 6 will result. In 
fact, for the pure pursuit, one has 


_Po_ / tan* -1 d/2 tan M+1 0/2 \ 

\ /i - 1 M + 1 / 


( 8 ) 


By graphing or tabulating, 6 is known implicitly for 
any t. 


3.2.5 Centrifugal Force and Isogees 

It is important to assess the centrifugal force to 
which the pursuer is subject in traversing pursuit 
courses. On one hand, the circle of curvature can be 
put to use as a replacement for a segment of the 
curve (over the time of flight of a bullet from a de¬ 
fending bomber) in deriving approximate deflection 
formulas. On the other hand, a knowledge of centrif¬ 
ugal load leads to an estimate of the angle of attack d 
of the fighter and its effect on the course flown. 
Finally this force or load gives the boundaries in re¬ 
gard to range and angle that a fighter may reach 
before physiological or structural limitations become 
operative. 

For a pure pursuit curve, the radius of curvature R 
is given by R = vpdt/dd, since the tangent to the 
circle of curvature is also the line to which the angle 

d The angle of attack is the angle between the direction of 
motion of the aircraft and some reference line (such as a wing 
chord or a gun’s bore axis or a longitudinal axis) that lies in 
the plane of symmetry of the aircraft. 


CONFIDENTIAL 




















36 


PURSUIT CURVES 


off 0 is measured. The centrifugal load n c in units 
of gravity is found from 

v F v B sin 0 ... 

n c - - (9) 

gp 

If n c is given successive constant values, then the 
curves of equal load ( isogees ) that arise are, clearly, 
circles tangent to the pursued’s track and of radius 
VFVB/2gn c . If such a family of isogees is drawn and if 
a family of pursuit curves is superposed, as in Figure 
6, one may readily find the load for any curve at any 


90 ° 



YARDS 

Figure 6. Isogees and pursuit curves. 


point. As a sample conclusion it is immediately evi¬ 
dent that with high-speed aircraft close approach at 
any material angle off the bomber’s tail is prohibited 
by the high loading. This can be adduced as one 
argument for dispensing with all armament other 
than that of the defending tail of an ultra high-speed 
bomber. 221 With such a double family one could also 
trace out the growth and decay of load. The maxi¬ 
mum load, when such exists, is found with ease 
analytically by maximizing n c . The simple result is 
that n c max occurs at an angle off determined by 


This locus appears as a dotted radial line in 
Figure 6. 

It is possible, in view of equation (9), for the fighter 
to reduce the load to which he is subject by deliber¬ 


ately using a slow speed during the firing run. As the 
fighter’s speed decreases the maximum load dimin¬ 
ishes steadily. e It may be presumed that the difficulty 
of the fighter’s aiming problem is reduced, and that 
he may do his firing at greater angles off the bomber’s 
stern, which is equivalent to offering a more difficult 
shot for the defense because of the higher angular 
rate. In addition, a longer period of sustained fire is 
available. 10 - 129 The tactical disadvantage is that the 
fighter will close range up to a certain point and then 
will fall back. 

Since a lead pursuit curve is usually a better ap¬ 
proximation to the curve actually flown than is a 
pure pursuit curve, it is sometimes desirable to calcu¬ 
late the centrifugal load for the lead pursuit. This 
formula, which is a companion of equation (9), is 

VfVb ,i . . . cos0\ nn 

n c = -(1 — pv) sin 0(1 — 1 '-- j* (II) 

gp \ cos 5 / 


3.2.6 Total Load Factor 


For use in the next section, which calculates a more 
realistic deviation function, the total load on an air¬ 
craft is required. Evidently, this total load n is a 
suitable vector sum of centrifugal force and that 
component of the gravitational force lying in a plane 
perpendicular to the direction of motion. Forces are 
summed in this way because the lift, which must 
support the effectively heavier aircraft, lies in that 
plane. In the formulas given below for total load n 
and bank angle R (roll), the turn angle Y (yaw) is 
measured in a horizontal plane after projection of the 
flight path, and the dive angle P (pitch) is measured 
in a vertical plane from the horizontal projection 
down to the flight path. It is assumed that the air¬ 
craft is flying cleanly — with no slip or skid — at a 
speed v. Then 130 

dY 
v —— 

tan R = -^> (12) 

g + v sec P — 


e It is a curious fact that if the fighter's speed is held con¬ 
stant and the bomber's speed is changed, there is a certain 
bomber speed which yields a least maximum load for the 
fighter. 2210 Indeed, if n c max is expressed as a function of /z by 
combining equations (10), (9), and (4), then a true minimum 
is found to occur at the single root of 




This root is 1.3. 


CONFIDENTIAL 












THE EFFECT OF ANGLE OF ATTACK ON PURSUIT CURVES 


37 


and 


cos P + 


vdP 
g dt 


cos R 


(13) 


in units of gravity. 

For special flight paths, equations (12) and (13) 
reduce to the following forms, which are useful in 
various computations: 

1. Circle of radius r in a horizontal plane. 

v 2 1 / 

tan R = — w = / i + 
gr \ 

2. Circle of radius r in a vertical plane. 



v 2 

tan R = 0 n — cos P + — • 
gr 

3. Circle of radius r in a plane of action of incli¬ 
nation CO. 


tan R = 


COS co 


V 2 

- b cos 6 sin co 

gr 


— j/cos 2 co + + cos 0 sin co ^ 


where 0 is the angle between the direction of flight 
and a horizontal line in the plane of action. 

4. Helix with horizontal axis and sinusoidally 
varying speed. (This is the case of corkscrew avoid¬ 
ing action by a bomber. The details 130 are compli¬ 
cated but straightforward and will not be set down.) 


3.3 THE EFFECT OF ANGLE OF 
ATTACK ON PURSUIT CURVES 

3.3.1 Deviation Function and 
Trajectory Shift 

In Section 3.1.2 it was pointed out that a knowl¬ 
edge of the exact curve flown by a real and perfect 
fighter permits the defending fire control to calculate 
future positions of the target. Since these future posi¬ 
tions depend intimately on the deviation 8 of the 
fighter’s velocity from the gun-target line (Figure 1) 
it is necessary to analyze 8 more carefully than was 
done in Section 3.2.2 [deviation (3)]. 

As footnoted in Section 3.2.5 (footnote d), an 
angle a exists between the bore axis of a fighter’s gun 
and the direction of motion. This angle of attack 
consists of a fixed and a variable part. The fixed part 
is attributable to the installational setting which 


allows for gravity drop. The variable part is caused 
by a change in the load factor n (Section 3.2.6), 
which requires a change in the angle of attack of the 
wings to supply a balancing lift for this change in 
aircraft weight. It follows that the direction of de¬ 
parture of the fighter’s bullet is along the diagonal 
of a parallelogram determined by the propellant 
muzzle velocity v 0 and the fighter velocity Vf- In 



Figuke 7. The effect of angle of attack of the bore axis. 


Figure 7, it is assumed that a lies entirely in the 
plane of action. (This it will not do, in general, be¬ 
cause the aircraft is banked.) The deflection problem 
is solved by equation (1) of Chapter 2, which gives 

A = — sin 6 H-— <*. (14) 

u Vo + Vf 

In Section 3.2.2 [deviation (3)] only the first term on 
the right of equation (14), the normal lead, was used 
for the deviation. The correct deviation function is 

8 = A — a. 

The pursuit curve generated by this 8 is called an 
aerodynamic lead pursuit. The next problem is to 
explore a. 

Before leaving equation (14), however, the mean¬ 
ing of the equation from the point of view of the 
fighter pilot is given by Figure 8. In this representa¬ 
tion, it is not assumed that a lies in the plane of 
action. Instead, the aiming allowance required by 
the second member on the right of equation (14) 
lies in the plane of symmetry of the fighter. It is 
called the trajectory shift . 190 Its size is also de¬ 
termined by an exploration of a. 

3.3.2 Angle of Attack in Terms of 
Load and Indicated Airspeed 

The resultant of all pressures on an aircraft wing — 
of the lower than atmospheric pressures on the upper 


CONFIDENTIAL 











38 


PURSUIT CURVES 


surface and of the equal to or slightly greater than 
atmospheric pressure on the lower surface — is re¬ 
solved into a lift, perpendicular to the direction of 
motion, and a drag, parallel to the direction of mo¬ 
tion. If the bullet in Figure 5 of Chapter 1 is replaced 
by a wing profile, that diagram illustrates this situa¬ 
tion also. The yaw of the bullet is equivalent to the 
angle of attack of the wing. The lift f L, in pounds, 
is given by 

L = Cl~^t Sv 2 i (15) 

z 

where Cl = lift coefficient (dimensionless), 

p = air density (slugs per cubic foot), 8 
S = wing area (square feet), 
v = true airspeed (feet per second). 11 


f The drag is given by a similar expression: 

D = Cd-^ Sv 2 (See Section 1.2.2.) 

'The aerodynamic air density p is an NACA standard. 
It differs from the ballistic standard p G , and must not be 
confused with the relative ballistic air density which is also 
denoted by p (Section 1.2.3). Both p (NACA) and p a (bal¬ 
listic) vary at a given altitude as the temperature and hu¬ 
midity change. 


Ballistic and NACA altitudes for given ballistic 
relative air density. 


Relative 

Ballistic altitude 

NACA altitude 

air density 

(feet) 

(feet) 

1.0 

0 

621 

0.8 

7,065 

8,015 

0.6 

16,175 

16,994 

0.4 

29,013 

28,661 

0.2 

50,960 

44,592 

Standard atmosphere 

based on NACA 

Report 

No. 218. 



Altitude (feet) 

Po/p 

V Po/p 

0 

1.000 

1.000 

2,000 

1.061 

1.030 

5,000 

1.161 

1.077 

10,000 

1.354 

1.164 

15,000 

1.590 

1.260 

20,000 

1.877 

1.370 

30,000 

2.675 

1.636 

40,000 

4.086 

2.021 


h An airspeed meter measures V 2 p ( TAS ) 2 , where TAS is the 
true airspeed. It is calibrated to read TAS at sea level (p = p 0 ). 
Hence if IAS is the indicated airspeed, 



3 The additional superelevation allowance for gravity drop 
is about one-fifth of L 2 . 


Experiment and theory show that the lift coeffi¬ 
cient is an almost linear function of the angle of 
attack of the wing chord over a range almost up to 
stall. Since the guns are installed at a fixed angle with 
respect to the wing chord to which angle of attack 



Figure 8. View through fighter sight of total deflec¬ 
tion. 

is usually measured, we may take the angle of attack 
a of the mean bore axis of the fighter’s battery in the 
form 

Cl = K\ol -f" K 2 , 

where Ki and K 2 are constants. 

If an aircraft is subjected to a load factor n , we 
must have L = Wn, where W is the normal weight 
of the aircraft. Consequently, 


Lin 


-L 2 


(IASy 

where Li and L 2 are constants given by 


Li = 


2W_ 

Kip 0 S 


L _£* 

U ~K, 


(16) 


By an aerodynamic argument 57>117 which will not 
be reproduced, L x may be calculated by the formula 



where S is the wing area in square feet and b is the 
wing span in feet. When equation (17) is used in 
equation (16), a will be in radians, IAS in miles per 
hour, and W in pounds. Next, L 2 depends on the 
installational angle of the guns. If we suppose 31 that 
the guns are adjusted so that at a particular level 
flight speed, (IAS) 0 , they are horizontal, 1 then 

4 1 = ( IAS )l- (18) 


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THE EFFECT OF ANGLE OF ATTACK ON PURSUIT CURVES 


39 


The value assigned to (IAS)o has frequently been 
250 mph. Using this value and standard airplane 
dimensions, average values of Li and L 2 are given by 
Table 1 for certain fighters of World War II. 


Table 1 . Angle of attack constants for World War II 
fighters. 


Class 

Types 

U 

U 


Mel09, MellOG, 

FW190A, Ju88C 

3,500 

0.056 

G , 

Me209, Me210, Me410 

4,700 

0.075 

Ji 

Zeke, Hamp, 

Oscar I and II 

2,067 

0.033 

Ji 

Tojo, Nick, Tony, 

Average American 

3,100 

0.050 


3.3.3 Vicious Circle of This 
Approach 

Formula (16) requires a knowledge of n, and to de¬ 
termine the load the curvature of the path must be 
known. A vicious circle is completed since the path 
cannot be determined until 8 (and so a , and so n) is 
known. The correct resolution of this difficulty is 
given in Section 3.4.2. In the early literature, 29 the 
centrifugal load for a pure pursuit or a lead pursuit, 
i.e., equations (9) or (11) or some average of these, 
at the point in question, has been used. From the 
practical point of view, refinement at this point is 
somewhat absurd since the exact speed of the at¬ 
tacker cannot be known, and centrifugal load is always 
proportional to this speed. The only justification is 
found in a desire to avoid bias in calculations which 
are to be used to meet an average tactical situation 
rather than a particular one. 

With such an approximation for n and under the 
assumption of constant speed, it is possible to inte¬ 
grate the differential equations (1) and (2), using a 8 
determined by equations (14) and (16), by numerical 
methods. This will not be done since defensive gun¬ 
nery is interested only in a segment corresponding to 
the time of flight of the bomber's bullet. The required 
deflection can, in fact, be made out as a function of 
range and angle off of a fighter, rather than as a 
function of position along a particular curve. (See 
Section 4.2.2.) 

Further analysis of aerodynamic lead pursuit 
curves by these methods has treated 170 the com¬ 
ponents of a in and at right angles to the plane of 
action. The perpendicular component causes the air¬ 


craft to move slightly below the instantaneous plane 
of action (or sag). 

The patchwork theory of this section is correctly 
revised in Section 3.4. The detail supplied here has 
intrinsic value and is an exemplar of approximation 
methods in this field. 


3.3.4 Qualitative Effect of Angle 
of Attack 

The effect of angle of attack on a pursuit curve 
may be summarized qualitatively by giving the rela¬ 
tions among pure, lead, and aerodynamic lead pursuit 
curves that originate at the same point and have the 
same v B and vf : (1) at long ranges, the load is low 
(close to 1), the angle of attack is small, and the 
aerodynamic curve almost coincides with the lead 
pursuit; (2) as the range closes, the load generally in¬ 
creases, the angle of attack increases, and, since the 
shifted trajectory must lead to impact, the fighter's 

(1) LOW LOAD 

(2) HIGH LOAD 

(3) ULTRA-HIGH LOAD 

FIGHTER 0 'S THE SAME FOR ALLCASES 



Figure 9. Effect of load on fighter’s direction of 
motion. 


velocity is directed more toward the bomber than it 
would be for a lead pursuit, so that the aerodynamic 
curve edges over toward the pure pursuit; and (3) for 
very high loads, the angle of attack may be so great 
that the fighter's velocity is directed behind the 
bomber, and yet the shifted trajectory leads the 
bomber by an amount sufficient to cause a hit. These 
three facts are partially illustrated by Figure 9. 


CONFIDENTIAL 









40 


PURSUIT CURVES 


3.4 TRUE AERODYNAMIC LEAD 

PURSUIT CURVE 

3.4.1 New Variables — Fighter 

Speed, Course Curvature 

In Section 3.3 the real analytic problem was side¬ 
stepped. Of the various approximations made in that 
section those regarding the load factor and constant 
speed on the part of the attacking fighter are crucial. 
Both do violence to the real dynamics of the situ¬ 
ation. In the first instance, given an unknown curve, 
its rate of change of direction should enter naturally 
as an unknown. In the second instance the aircraft 
should be permitted to change speed naturally under 
the forces of thrust, drag, and gravity. Consequently, 
speed should also be a variable of the description. 

The improvements to be made may be recognized 
most readily by considering the simple case of an 
attack, made in a vertical plane, 24 by a fighter on a 
bomber which moves on a straight and level track at 
constant speed. If a vertical plane is used, trigono¬ 
metric details do not obscure the new approach. The 
new ideas are also kept clear by assuming that the 
bore axis of the fighter’s gun is kept on the target, 
so that deflectional and ballistic notions are not 
present. 


LIFT, L 



Figure 10. A typical instant of true aerodynamic 
pursuit. 


Figure 10 shows the force system j on the airplane, 
and indicates that a further simplifying assumption 

J Aerodynamic forces acting on the tail surfaces of the 
fighter are neglected since the gun is kept on target by as¬ 
sumption and a description of how this is done by elevator 
moments is not required. 


— the bore axis and thrust axis coincide — has been 
made. The tangential and normal dynamical equa- 


tions are respectively 



W dv F . 



— — = W sm P + T cos a — D, 
g dt 

(19) 

and 


Wv[ 

W dP 


9 R 

— - v F — = L — W cos P + T sin a, 

9 dt 

(20) 


where R is the radius of curvature. The kinematical 
equations of pursuit are 


dp 

— = — Vf cos a + v B cos ( P — a), (21) 

dP da 1 

~ = - ~ [v F sin a + v B sin (P - a)]. (22) 

To utilize this set of nonlinear equations it is neces¬ 
sary: to determine the aerodynamic constants for a 
particular airplane at a particular throttle setting; 
to assign suitable initial values to the variables, v Ff 
P , p) and to proceed with the integration by a 
systematic process. Definite formulas are available, 24 
enabling one to proceed directly from the perform¬ 
ance values of propeller efficiency, maximum engine 
brake power, and the corresponding maximum level 
flight speed at a certain altitude, when combined 
with the weight and airplane geometry, to the con¬ 
stants T, D, L of equations (19) and (20). Turning 
next to the initial values, v F , P, and p may be speci¬ 
fied at pleasure at t = 0. 

It is not at all evident, however, what initial value 
for the angle of attack a should be selected. This 
choice is connected with the nature of that part of 
the flight path to which some license is permitted, 
which just precedes initiation of the guns-bearing 
phase. Thus, depending on the choice of a in the 
usual range from 2° to 12°, it would appear that a 
family of curves is possible. Fortunately, however, 
numerical integration demonstrates the existence of 
a boundary layer effect. That is, the dive angle versus 
time curve will show a sharp hook over an interval of 
about one-half second. This means that different 
initial choices of a generate a funnel leading to a 
unique value of P and continuing as a single curve. 
By extrapolation of this single curve back to the 
initial time, what may be called a natural initial value 
for a is determined. As the final point in the discus¬ 
sion of the set of equations, it may be noted that al¬ 
though numerical integration is feasible a judicious 
blend 25 of graphical and nomographic procedures is 
enlightening. 


CONFIDENTIAL 









TRUE AERODYNAMIC LEAD PURSUIT CURVE 


41 


3.4.2 Complete Three-Dimensional 
Equations 

The discussion of the true aerodynamic lead pur¬ 
suit curve of Section 3.4.1 may now be extended to 
three dimensions. The treatment is complicated by 
the banking of the fighter and by the introduction of 
ballistic considerations in determining the lead taken 
by a perfect fighter pilot at each instant. (A final 
complication, that of allowing for sideslip by means 
of an additional term involving a cross force, has 
been considered in one account. 26 This leads to more 
variables than equations — because of the freedom 
permitted the pilot in the amount of slip — and 
does not furnish a unique curve. There is experi¬ 
mental evidence 39 > 181 that fighter pilots can fly 
courses cleanly.) It will be assumed below that there 
. is no sideslip. In addition to this stipulation, matters 
are further qualified by supposing that the bomber 
proceeds on a straight and level track at constant 
speed, that the fighter’s throttle setting is left un¬ 
changed, and that the effect of gravity on the fighter’s 
bullets may be neglected. The last assumption is 
tenable because the effect of gravity on time of flight 
is negligible and the drop itself, over the ranges in 
question, is sensibly removed by a slight elevation of 
the gun over the sight line. Bullet patterns are not 
considered. It is supposed that variable aim is taken 
to produce continuous bullet impact at one point of 
the bomber target. 

The most convenient set of equations that has 
been derived 39a is set up with respect to the recti¬ 
linear trajectory traversed in space by the bullet in 
going from the fighter to the point of impact. This 
air range is denoted by r and has an azimuth 0 
measured counterclockwise from the forward direc¬ 
tion of bomber travel at the point of impact and an 
elevation 9 with respect to the horizontal plane. The 
angle a denotes the angle of attack of this trajectory, 
i.e., the angle between the direction of motion of the 
fighter and the trajectory at the moment of de¬ 
parture. The angle ei is the angle fixed at installation, 
from thrust axis to gun. The bank angle ft of the 
fighter is the angle from that perpendicular to the 
trajectory that lies in a vertical plane to the per¬ 
pendicular to the trajectory that lies in the fighter’s 
plane of symmetry. 

The ballistic elements present are: v Q , the muz¬ 
zle velocity of the fighter’s bullet; u 0 , the velocity 
of departure of that bullet; (v 0 -\r vf)', and 
b = 0.00186p/c 5 where p is the relative ballistic air 


density and c 5 is the appropriate ballistic coefficient. 
In addition to these primary terms, two secondary 
combinations may be introduced. The angle ai = 
(vq vf/vq) a — ei represents physically the angle 
from thrust axis to the aircraft’s direction of flight. 
Again, in units of feet and feet per second, the time 
of flight tf is given by equation (3) of Chapter 1 and is 

r 

\/uq 

f ~Vu 0 -br' 

The derivative of this with respect to time is simply 


tf ( Vu 0 -br ) 2 

With these letters the tangential equation is 
= W (cos a sin 9 -f- sin a cos 9 cos ft) 

g dt 

T - T cos <x\ — Z). (23) 


It is evident that this summing of forces in the flight 
direction gives an equation of the same form as 
equation (19). Next, because two angles are required 
to specify r, two equations are required in the normal 
directions. These equations, taken together, form an 
analoguefor the two-dimensional case [equation (20)]. 
The first equation is 

W ( . d4> dd da\ 

- — tM sinft cos0— + cosft— + — J 

g \ dt dt dt / 

= L — W( cos a cos 9 cos ft — sin a sin 9) + Tsinai. 

(24) 

The structure is clear and the projection factors (in 
parentheses) are exposed. The third equation con¬ 
tains only accelerational and gravitational forces and 
will therefore be homogeneous in W. It is 


w r 

— V F (sin 9 — cot a 
Q L 


cos 9 cos ft) 


+ cot a sin ft 


sin 


dftl 

a — 

dt J 


d(f> 

dt 


W sin ft cos 9. (25) 


We must expect to find three kinematic equations 
also. The required deviation function is found in¬ 
directly by noting that the distance from the position 
of the bomber at the moment of fire to the future 
impact point along the bomber’s track is v B tf where 
tf is the time of flight over r. It follows that the speed 
of the point of impact is v B + v B t f . It is such a ghost 
point which is being pursued since all work is relative 
to the trajectory. Consequently by projection of the 


CONFIDENTIAL 





42 


PURSUIT CURVES 


fighter speed and of the ghost speed on the trajectory 
the range rate equation 
dv 

— — — Vp cos a — v B {\. + tf) cos (f> cos 0 (26) 

dt 

arises. Finally, two equations for the rates of change 
of azimuth and elevation of the trajectory are written 
down via projections normal to r. These equations are 

— =--— [vf sin a sin ft — v B (\ + tf) sin <£] (27) 

dt r cos 0 

and 

— = — — [vf sin a cos ft — v B {\ +£/)cos<£sin0]- (28) 

dt r 

It will be remembered that r will occur in the left- 
hand members of the last three equations through t /. 



0 50 100 150 200 250 300 350 


p SIN 0 (YARDS) 

Figure 11. Comparison of pursuit curves. 

Instead of azimuth and elevation of the trajectory 
it would be possible to use as angular coordinates the 
angle off bomber’s track of the trajectory and the 
elevation angle of the (instantaneous) plane of ac¬ 
tion. This would be an heuristic rather than a 
technical improvement. It would bring the situation 
in symmetry with the usual pursuit curve description 
in terms of range and angle off, and would expose 
sag (Section 3.3.3) by showing the slow rotation of 
the plane of action. 

It is also evident that by proper choice of the func¬ 
tion giving distance between bomber and the ghost 


point, arbitrary behavior on the part of the attack¬ 
ing pilot, rather than perfect lead behavior, can be 
introduced painlessly. 

Systematic and precise methods of numerical inte¬ 
gration of such systems are available, 3915 and a con¬ 
siderable number of courses covering modern tactical 
ranges including high-speed bombers and fighters 
has been computed. 40 (An example of a true aero¬ 
dynamic lead pursuit is given in Figure 11.) Com¬ 
puted courses have been carefully compared with 
those actually flown. 390 During a sight assessment 
program at the Patuxent River Naval Air Station, 
an F6F-3 fighter equipped with a Mark 23 gyro gun 
sight was flown in pursuit of a bomber whose speed 
was 130 knots at an altitude of 6,000 ft. From camera 
records the range, azimuth, and elevation of the 
fighter with respect to the bomber, and the fighter’s 
lead, at quarter second intervals, were known. To 
test the theory, it is reasonable to select those at¬ 
tacks in which the fighter’s lead was well-nigh per¬ 
fect. When this is done the coincidence between 
computed and observed values is quite remarkable, 
showing an average absolute range difference of 
about 9 yd, and average absolute differences in angle 
of about 10 mils. 

In concluding this section, it is to be emphasized 
that the rounded character of this theory should 
make it immediately applicable to future problems 
in aerial warfare whether they deal with aircraft or 
missiles. 

3.5 CERTAIN TACTICAL CONSIDERATIONS 

3.5.1 Combat Maneuvers by Bomber 

Throughout this chapter, the bomber under pur¬ 
suit has moved on a straight line course at constant 
speed. This rectilinear assumption is tenable in the 
light of the requirements of massive formations, the 
nature of bombing runs, and consideration of range 
of operation. However, whenever a bomber must act 
alone or in a very small formation, as may well 
happen at night, on patrol, or on pathfinding tours, it 
may suffer saturation and coordination attack by 
enemy fighters, against which support fire power 
cannot be brought to bear. Violent and frantic eva¬ 
sive action suggests itself as a defensive measure. 
But further thought shows that although such ran¬ 
dom course changes may make the aiming problem 
of an attacking fighter more difficult and may de¬ 
stroy the coordination of two or more attackers, it 


CONFIDENTIAL 























CERTAIN TACTICAL CONSIDERATIONS 


43 


will at the same time deceive the gunners of the 
bomber itself. Instead of evasive action the concept 
of combat maneuvers arises. This is a deliberate, 
planned, and properly timed maneuver designed to 
reduce the possibility of damage to the bomber 182 by 

(1) offering the fighter a changing deflection in 
amount and in line, (2) shortening the attack, (3) in¬ 
creasing the loading on the fighter, and (4) making 
use of a violent and turbulent slipstream. Since the 
maneuver is planned and practiced, one may pro¬ 
vide defending gunners with a specific set of simple 
rules for return fire, 178 ’ 192 which may therefore be 
reasonably effective. The most common combat 
maneuvers are steep diving turns and corkscrews. 
Only the corkscrew will be discussed. This maneuver 
has the added advantages of maintaining mean track 
and (sensibly) height, and of being the best counter 
against a coordinated attack. 

It is not profitable to attempt an exact mathe¬ 
matical solution of the problem of a fighter following 
a bomber in the corkscrew. The fighter cannot lay 
off deflection nearly as well as he may for the recti¬ 
linear case. This means that he will point in a random 
fashion in the neighborhood of the lead point, as air 
experiments show. 193 When necessary, one may 
rapidly appraise the situation by an obvious three- 
dimensional analogue of the midpoint method of 
Section 3.2.4. A standard corkscrew to port may be 
described as follows: 182,192 (1) diving turn to port, 
changing course about 30°, losing perhaps 1,000 ft, 
and building indicated airspeed up to 220 mph, 

(2) climbing turn to port gaining perhaps 700 ft, and 
losing speed to 180 mph, (3) rolling and continuing 
with a climbing turn to starboard gaining perhaps 
300 ft, and losing speed to 150 mph, (4) diving turn 
to starboard until speed is about 190 mph, (5) rolling 
and continuing with a diving turn to port until 
speed is about 220 mph. And the cycle may be con¬ 
tinued. Note that after item (3) the course should be 
back on the opposite side of the original heading by 
30°. About 500 ft will be lost during the cycle, and 
50° will be about the maximum bank. The time per 
cycle is about 47 sec, which allows 4 sec for each 
change and for each roll, and allows 8, 9, 5, 9 sec 
respectively for the four phases of a cycle. 

3.5.2 Effect of High Mach Numbers 
on an Attack 

The trend in military aviation is toward higher and 
higher speeds and altitudes for both bombardment, 


and, consequently, pursuit operations. When a high¬ 
speed fighter attempts to attack a high-speed bomber 
on a pursuit curve, it is apparent from equation (9) 
that high loading will result at material angles off 
stern, and at those ranges within which ordinary 
ammunition is effective. Although it is not known 
exactly how load and growth of load affect the pilot's 
aiming, there still exist upper load bounds which may 
not be exceeded for physiological and aerodynamical 
reasons. Blackout (but not the weight of the limbs) 
may be inhibited by the use of suitable antipressure 
suits. The aerodynamical restriction, however, being 
a question of wing design, is not so readily dismissed. 
This restriction may be considered in some detail. 119 



0.2 0.4 0.6 0.8 1.0 

MACH NUMBER,M 


Figure 12. Typical critical lift coefficient curve. 

The Mach number M is the ratio of the speed of a 
body to the speed of sound. Since the speed of sound 
decreases with altitude from about 764 mph at sea 
level to about 666 mph at 50,000 ft (true airspeed), 
those modern aircraft designed to perform best at 
altitude must accept Mach numbers approaching 
unity. For each wing there exists a curve giving a 
critical value for the lift coefficient Cl in terms of the 
Mach number. At any given Mach number, an at¬ 
tempt to exceed this critical value will lead to ex¬ 
cessive shudder and vibration which may result in 


CONFIDENTIAL 












44 


PURSUIT CURVES 


structural damaged Consequently, if the effective 
weight Wn of the airplane requires a lift coefficient 
in excess of C^crit, shuddering flight must be accepted. 
Figure 12 shows a typical curve. It resembles the 
critical curve for jet fighters. 

With the aid of the C L crit curve, regions of com¬ 
fortable flight in coordinates of load versus indicated 
airspeed may be obtained. The IAS of stall at 
M = 0.2 is an efficient parameter in the discussion. 
A readily deduced approximate formula for the 
stalling speed at M = 0.2 is 

Voo = 16 \Aving loading, 

where is in miles per hour, and the loading is in 
pounds per square foot. With the aid of equation (15), 
the curve of critical load, n, versus IAS, v, for a given 
altitude, is given by 

C L crit ~ v 2 

^ C /A 9 1 ! II 2 F(v), 

U L crit ^00 

where a is the IAS of sound at the chosen altitude. 
An F(v) family, with the altitude as parameter, is a 
nested set of arches with legs set on the line n — 1. 
As the altitude increases, the region of comfortable 
flight so bounded shrinks materially. Such curves, 
when used in conjunction with the load factor formu- 

k For very low Mach numbers, an attempt to exceed Cl crit 
leads to the usual type of stall. 


las of Section 3.2.5 and the third form of Section 
3.2.6, define a region, bounded by a closed range 
versus angle off curve, within which a fighter cannot 
penetrate on a pursuit curve attack. 

3.6 SUMMARY 

Section 3.1 discusses the tactical setting of pursuit 
curves in aerial warfare, and considers the reasons 
why a detailed study of such curves is a prerequisite 
for gunnery investigations. 

Section 3.2 deals with the geometry of pure and 
deviated pursuit curves. The centrifugal forces ex¬ 
perienced in such curves is computed, and general 
formulas for the total force — gravity plus centrif¬ 
ugal — are given. 

Section 3.3 reviews and refines the theory of fixed 
gunnery in order to determine more exactly the angle 
by which the attacking fighter deviates from pure 
pursuit. The angle of attack of the fighter’s guns is 
explored in detail, since this determines the deviation 
as much as does the deflection taken by the fighter. 

In Section 3.4, the anatomy of the true aerody¬ 
namic lead pursuit curve is considered. The impor¬ 
tant points are that the fighter speed and the curva¬ 
ture of the course are now variables of the problem. 

Section 3.5 discusses avoiding action on the part 
of the bomber and the aerodynamic limitations on 
the fighter in making a pursuit curve attack. 


CONFIDENTIAL 






Chapter 4 


OWN-SPEED SIGHTS 


4.1 INTRODUCTION 

4.1.1 Positional and Rate Deflection 

Formulas 

T he deflection formulas deduced in Chapter 2 
are essentially of two types. One type uses 
velocities and angles to express the lead and the 
other employs angular rates measured at the gun. 
Since the first type is sensibly positional, simple 
mechanization appears plausible. The more compli¬ 
cated problems introduced by the necessity of meas¬ 
uring rates will be dealt with in Chapter 5. This 
chapter considers only developments of the non- 
angular-rate formulas. 

4.1.2 Prediction of Approach Angle 

It was pointed out in Section 2.2.3 that positional 
formulas, of which equation (1) is typical, require a 
value for the approach angle of the target, and that 
this value is not readily measured. The alternative is 
to predict the approach angle. This can be done by 
assuming that a quite special tactical situation is at 
hand. The argument follows. 

The primary function of the defensive gunnery of 
a bomber is to prevent its own aircraft from being 
shot down. Supporting other members of a formation 
by cross fire is a secondary function, and decimating 
an enemy fighter force is tertiary. A fighter attacking 
on a pursuit curve is most likely to shoot down the 
bomber, so that the defensive gunnery — in exer¬ 
cising its primary function — must deal adequately 
with such attacks. Fortunately, approach angle and 
course curvature can be predicted for pursuit curves, 
and this prediction may be utilized in the design of 
a mechanism. By taking advantage of such a special 
and dangerous situation, simple fire control may be 
developed which may be operated with a minimum 
of manipulation error on the part of the gunner. 
This is the basic logic of the chapter. 

It is obvious that fire control should commit itself 


to such prediction of the attacker’s behavior only if 
warranted by the current tactical situation and, even 
then, only if more flexible control is less effective be¬ 
cause of difficulties in design, production, or manipu¬ 
lation. Since, in addition, such commitment implies 
that the defense lags the offense (it permits the op¬ 
ponent to set the tactics before it can be designed), it 
follows that this type of dependence on enemy 
cooperation is a stopgap procedure. This chapter is 
the history of such a stopgap — the own-speed sight. 

A specialized weapon is designed to meet an ex¬ 
pected average of a given class of tactical situations, 
e.g., average fighter speeds and average attack 
ranges. A proponent of a more flexible weapon, indi¬ 
vidualized to account for variations in the opponent’s 
behavior, may justly seize upon this point. 

4.1.3 Definition of Own-Speed Sight 

An own-speed sight is a mechanism that displaces 
the bore axis of a gun from the line of sight by an 
angle v G sin t/v 0 , where the gun-mount velocity v G 
and the muzzle velocity v Q are preset. The variable 
size of this angle depends, then, only on the positional 
angle r, or the angle off of the target with respect to 
the aircraft’s direction of motion. The displacement 
angle is laid off toward the rear of the firing aircraft in 
the plane of action. Since mount velocity and pro¬ 
pellant velocity combine according to such an angle 
in yielding the bullet’s velocity of departure, a bullet 
directed by this mechanism will depart along the 
line of sight. To hit a fixed ground target on a calm 
day, the gunner need only hold the sight on target 
and fire. There is no ranging with an own-speed sight. 
To hit a fighter attacking on a pursuit curve, some 
percentage of v G (generally less than 100) is set in to 
decrease the full own-speed deflection. The reason is 
that the fighter deviates forward of the line of sight 
(of the instant of fire) during the bullet’s time of 
flight. Much of this chapter is concerned with this 
percentage factor. 


CONFIDENTIAL 


45 


48 


OWN-SPEED SIGHTS 


restricted cone of fire. It also shows that no real gain 
in accuracy would result by elaborating the mecha¬ 
nism to vary the percentage as some function of angle 
off. However, it is clear that variation with altitude 
and fighter speed is significant and should be taken 
into account. 


Table 1. Average percentages of own-speed deflec¬ 
tion. 


Fighter class 
Fighter 
speed (TAS) 

Gi 

300 400 

g 2 

300 400 

Ji 

300 400 

J 2 

300400 

Altitude 






0 

70 

61 

71 62 

68 61 

69 61 

10,000 

77 

68 

80 70 

74 67 

76 68 

20,000 

85 

75 

89 78 

80 73 

84 75 

30,000 

92 

82 

97 85 

86 78 

91 81 

40,000 

100 

89 

106 93 

92 84 

98 88 


Table 2. 
centages. 

Rules for modification 

of average per- 



Fighter class 


G\ 

G 2 J 1 J 2 


1. The percentage at 20,000 ft for 
350-mph fighters is 

80 

83 

76 

79 

2. For every 10,000-ft increase in 
altitude add 

7 

8 

6 

7 

3. For every 100-mph increase in 
fighter speed subtract 

9.5 

11 

8 

9 

4. For every 100-yd increase in 
beam range of 700 yd subtract 

3 

5 

2.5 

3 


A final reduction of the above tables has been 
suggested cooperatively by the Research Division of 
the Army Air Forces Central School for Flexible 
Gunnery and the Applied Mathematics Group at 
Columbia. 30 First decide on the type of fighter with 
which one must contend, e.g., class J 2 . Secondly, 
assume that this fighter will operate at essentially 
the same IAS at all altitudes, e.g., 275 mph. This 
will correspond to an increasing TAS at increasing 
altitude in accordance with the footnote to Section 
3.3.2. Thirdly, construct a chart such as that illus¬ 
trated in Figure 3, in which the slope of each line is 
the proper k 2 ~ l . In this chart the optimum percentage 
has been absorbed to produce a fictitious own speed 
Vq = k 2 v G , and full own-speed shooting can be done 
with v G . Finally, a navigator has available at all 
times his TAS and altitude. He is in a position to 
relay to gunners, via such a chart (and an intercom), 
the appropriate v G . 


4.3 VERIFICATION OF THEORY 

4.3.1 Pro and Con Combat Evidence 

The own-speed sight, with some optimum per¬ 
centage of the true airspeed of the gun mount set in, 
is useless against target paths which differ markedly 
from aerodynamic lead pursuit curves. In justifica¬ 
tion, then, of this specialized weapon, it must be 
established that pursuit curve attacks were the usual 
attacks during the period when this method of fire 



100 120 140 160 180 200 

GUNNER'S AIRSPEED (MPH) 

Figure 3. Calculation in the air of own-speed setting. 

control was to be used. It was German opinion c 
that 95 per cent of all attacks of World War II, made 
by German fighters during daylight on Allied bomb¬ 
ers, were pursuit curve attacks with an attempt at 
properly varying deflection. More objective evidence 
is supplied in Figure 4. A rather random selection d 
was made from a collection of 285 attacks (1944) by 
German fighters. The analysis was carried out by the 
assessment of combat gunnery film. Records of these 
attacks are also available which show the aim wander 
and plot angle off versus range. When hits are dis¬ 
tributed over such great range intervals, it is obvious 
that some facsimile of a pursuit curve is being flown. 

There are three parts to the counterevidence. 
First, the Japanese seemed to prefer the use of fixed 


c Interrogation of Th. W. Schmidt, Director of Film 
Analysis Section, by E. W. Paxson at Coburg, Germany, 
July 5 to 8, 1945. 

d Any bias in this selection is in the direction of choosing 
attacks with many hits as opposed to those in wFich bursts 
were fired throughout the attack with indifferent success. 
But even in the latter case the pilot w r as trying to fly lead 
pursuit. This represents a deviation from the average for 
which the fire control is designed. 


CONFIDENTIAL 


























VERIFICATION OF THEORY 


49 




FW 190 AGAINST 8* 17 
(REAR HEMISPHERE) 

RANGE IN METERS 


200 400 600 000 1000 1200 



























* 

























ME 109 AGAINST 8*17 
(REAR HEMISPHERE) 

0 200 400 600 800 1000 1200 


H-4 m t II II-1-1—44-4- 


0 


FW 190 AGAINST B* 17 
(FRONT HEMISPHERE) 

500 1000 1500 2000 























Figure 4. Bursts and hits within the burst as a function of range. 


medial deflection pursuit curve attacks. 188 It does not 
follow that own-speed sights will perform badly 
against such target paths. Second, against high-speed 
bombers (B-29 at a TAS of 300 mph at 24,000 ft), 
there is experimental evidence that the leads taken 
by a fighter executing a frontal attack will differ 58 
from those predicted by pursuit curve theory. A bias 
in performance of an own-speed sight would occur. 
Third, the closing months of World War II saw in¬ 
creased emphasis in both Europe and the Pacific on 
offset gun attacks. (See Chapter 8.) Again, saturation 
attacks from the rear by massed fighter formations 
(see Section 4.6) may require much support fire. 

One tends to conclude that the emphasis on de¬ 
fensive deflection of the own-speed type for positions 
such as waist, tail, and nose of moderate-speed 
bombers was not misplaced throughout most of 
World War II. 


4.3.2 Check of Optimum Percent¬ 
ages by Airborne Experiment 

At the request of the Army Air Forces Board, a 
check on the validity of the theory of optimum per¬ 
centages described in Sections 4.2.3 and 4.2.4 has 
been carried out. 59 Experimental data giving the 
bearing, range, speed, and deflection taken, for an 
attacking fighter, are available from careful camera 
analyses arising from assessment studies of lead com¬ 
puting sights. (See Chapter 7.) Since the path 
actually flown by the fighter was accurately known 
it was possible to compute the correct defensive de¬ 
flection at each instant. An experimental percentage, 
k 2 exp, was chosen which minimized the sum of the 
squares of the errors in the plane of action. On the 
other hand, to calculate an optimum percentage, 
k 2 opt, the correct fighter type, average speed, and 


CONFIDENTIAL 










































48 


OWN-SPEED SIGHTS 


restricted cone of fire. It also shows that no real gain 
in accuracy would result by elaborating the mecha¬ 
nism to vary the percentage as some function of angle 
off. However, it is clear that variation with altitude 
and fighter speed is significant and should be taken 
into account. 


Table 1. Average percentages of own-speed deflec¬ 
tion. 


Fighter class 
Fighter 
speed (TAS) 

G ! 

300 400 

g 2 

300 400 

J i 

300 400 

J 2 

300400 

Altitude 





0 

70 61 

71 62 

68 61 

69 61 

10,000 

77 68 

80 70 

74 67 

76 68 

20,000 

85 75 

89 78 

80 73 

84 75 

30,000 

92 82 

97 85 

86 78 

91 81 

40,000 

100 89 

106 93 

92 84 

98 88 


Table 2. Rules for modification of average per¬ 
centages. 


Fighter class 
G x G 2 J ! J 2 


1. The percentage at 20,000 ft for 
350-mph fighters is 

80 

83 

76 

79 

2. For every 10,000-ft increase in 
altitude add 

7 

8 

6 

7 

3. For every 100-mph increase in 
fighter speed subtract 

9.5 

11 

8 

9 

4. For every 100-yd increase in 
beam range of 700 yd subtract 

3 

5 

2.5 

3 


A final reduction of the above tables has been 
suggested cooperatively by the Research Division of 
the Army Air Forces Central School for Flexible 
Gunnery and the Applied Mathematics Group at 
Columbia. 30 First decide on the type of fighter with 
which one must contend, e.g., class J 2 . Secondly, 
assume that this fighter will operate at essentially 
the same IAS at all altitudes, e.g., 275 mph. This 
will correspond to an increasing TAS at increasing 
altitude in accordance with the footnote to Section 
3.3.2. Thirdly, construct a chart such as that illus¬ 
trated in Figure 3, in which the slope of each line is 
the proper fc 2 _1 . In this chart the optimum percentage 
has been absorbed to produce a fictitious own speed 
Vq = k 2 VG, and full own-speed shooting can be done 
with Vq. Finally, a navigator has available at all 
times his TAS and altitude. He is in a position to 
relay to gunners, via such a chart (and an intercom), 
the appropriate Vq. 


4.3 VERIFICATION OF THEORY 

4.3.1 Pro and Con Combat Evidence 

The own-speed sight, with some optimum per¬ 
centage of the true airspeed of the gun mount set in, 
is useless against target paths which differ markedly 
from aerodynamic lead pursuit curves. In justifica¬ 
tion, then, of this specialized weapon, it must be 
established that pursuit curve attacks were the usual 
attacks during the period when this method of fire 



too 120 140 160 180 200 


GUNNER’S AIRSPEED (MPH) 

Figure 3. Calculation in the air of own-speed setting. 

control was to be used. It was German opinion c 
that 95 per cent of all attacks of World War II, made 
by German fighters during daylight on Allied bomb¬ 
ers, were pursuit curve attacks with an attempt at 
properly varying deflection. More objective evidence 
is supplied in Figure 4. A rather random selection d 
was made from a collection of 285 attacks (1944) by 
German fighters. The analysis was carried out by the 
assessment of combat gunnery film. Records of these 
attacks are also available which show the aim wander 
and plot angle off versus range. When hits are dis¬ 
tributed over such great range intervals, it is obvious 
that some facsimile of a pursuit curve is being flown. 

There are three parts to the counterevidence. 
First, the Japanese seemed to prefer the use of fixed 


c Interrogation of Th. W. Schmidt, Director of Film 
Analysis Section, by E. W. Paxson at Coburg, Germany, 
July 5 to 8, 1945. 

d Any bias in this selection is in the direction of choosing 
attacks with many hits as opposed to those in which bursts 
were fired throughout the attack with indifferent success. 
But even in the latter case the pilot w^as trying to fly lead 
pursuit. This represents a deviation from the average for 
which the fire control is designed. 


CONFIDENTIAL 

























VERIFICATION OF THEORY 


49 


FW 190 AGAINST B* 17 
(REAR HEMISPHERE) 




RANGE IN METERS 

200 400 600 800 1000 1200 













11111 . 1 .. 














1 11 .... 1 ... .' 

























ME 109 AGAINST B*I7 
(REAR HEMISPHERE) 

0 200 400 600 800 























1000 


1200 


0 


FW 190 AGAINST B* 17 
(FRONT HEMISPHERE) 

500 1000 1500 2000 























Figure 4. Bursts and hits within the burst as a function of range. 


medial deflection pursuit curve attacks. 188 It does not 
follow that own-speed sights will perform badly 
against such target paths. Second, against high-speed 
bombers (B-29 at a TAS of 300 mph at 24,000 ft), 
there is experimental evidence that the leads taken 
by a fighter executing a frontal attack will differ 58 
from those predicted by pursuit curve theory. A bias 
in performance of an own-speed sight would occur. 
Third, the closing months of World War II saw in¬ 
creased emphasis in both Europe and the Pacific on 
offset gun attacks. (See Chapter 8.) Again, saturation 
attacks from the rear by massed fighter formations 
(see Section 4.6) may require much support fire. 

One tends to conclude that the emphasis on de¬ 
fensive deflection of the own-speed type for positions 
such as waist, tail, and nose of moderate-speed 
bombers was not misplaced throughout most of 
World War II. 


4.3.2 Check of Optimum Percent¬ 
ages by Airborne Experiment 

At the request of the Army Air Forces Board, a 
check on the validity of the theory of optimum per¬ 
centages described in Sections 4.2.3 and 4.2.4 has 
been carried out. 59 Experimental data giving the 
bearing, range, speed, and deflection taken, for an 
attacking fighter, are available from careful camera 
analyses arising from assessment studies of lead com¬ 
puting sights. (See Chapter 7.) Since the path 
actually flown by the fighter was accurately known 
it was possible to compute the correct defensive de¬ 
flection at each instant. An experimental percentage, 
A /2 exp, was chosen which minimized the sum of the 
squares of the errors in the plane of action. On the 
other hand, to calculate an optimum percentage, 
k 2 opt, the correct fighter type, average speed, and 


CONFIDENTIAL 











































50 


OWN-SPEED SIGHTS 


altitude only, were used. (Since the fighter used 
caliber 0.50 AP M2 or API M8 in taking deflection, 
Tables 1 and 2 do not apply, since they were based 
on typical German and Japanese 20-mm ammuni¬ 
tion.) Points at an interval of one-half second were 
taken. A point is an O point if the fighter’s lead 
gave a mean point of impact on its target (a sphere of 
radius 7 yd). A point is a B point if it comes before an 
0 point by 2 sec or less. All other points are N 
points. Ranges were up to 900 yd only. The results 
are given in Table 3. The agreement is satisfactory. 


altitude of 31,000 ft by a class J 2 fighter, on a bomber 
flying at 385 mph. The ammunition used by both 
fighter and bomber agrees with that used in cal¬ 
culating Table 1. By interpolation in that table, 
k 2 = 0.80. The corresponding deflection A = 168 sin 
r agrees with the exact values with an average error 
of about one milliradian. A systematic check of this 
nature is at hand, 30a but is academic compared with 
that of Section 4.3.2. The errors in the plane of action, 
when particular values of k 2 op t are used, are of the 
order of 2 milliradians. 


Table 3. Theoretical and experimental own-speed percentages. 


Source 

Data for attacks 

Type 
of point 

Number 
of points 

100&2 opt 

100&2 exp 

Average error in 
plane of action 
using hi opt 
(milliradians) 

AAFPGC 

Eglin Field 
(ST 2-44-120) 

7 rear quarter, 6 beam, 

7 nose, 3 strafing 
v G = 178 mph (B-24) 
i'f = 343 mph (P-63) 

Alt = 8,000 ft 
(averages) 

O 

25 

79.6 

76.3 

6.6 

O, B 

41 

79.6 

76.5 

6.2 

O, B, N 

O 

138 

9 

79.6 

81.8 

87.3 

70.9 

32.8 

8.1 

AAFPGC 

Eglin Field 
(ST 2-44-22) 

1 rear quarter, 

1 beam, 5 nose 
v G = 207 mph (B-17) 
vp = 343 mph (P-63) 

Alt = 8,000 ft 
(averages) 

O, B 

17 

81.8 

68.2 

12.4 

O, B, N 

O 

48 

61 

81.8 

84.5 

63.9 

83.9 

18.7 

6.0 

Patuxent River 
Naval Air Stn. 
(Mk 23-Mk 8 
assessment) 

16 beam and rear quarter 
v G = 150 mph (B-24) 
v F = 300 mph (F6F) 

Alt = 6,000 ft 
(averages) 

O, B 

83 

84.5 

83.5 

5.7 

O, B, N 

157 

84.5 

80.6 

7.3 


For the second source, the sparseness of correct lead 
points means that the theory is not expected to hold 
well. In the opposite direction the good agreement in 
the third class can be attributed to the use of a lead 
computing sight by the attacking fighter which led 
to excellent shooting on his part. 

4.3.3 Analytical Checks 

It is also possible to validate the approximate 
theory analytically. A large number of perfect aero¬ 
dynamic lead pursuit curves has been computed 40 as 
sketched in Section 3.4.2. For each of these the cor¬ 
rect lead to be taken in the plane of action is given. 
The only course that can be checked immediately 
without calculation is Course L\. This is a flat tail 
attack, made at an average speed of 415 mph at an 


4.3.4 Earlier Work 

Earlier efforts at validating the theory 60 ’ 202 are 
mentioned bibliographically to complete the picture 
of experimental evidence. This work compared 85 per 
cent of own-speed deflection with correct deflection 
and suffered from many crudities. 

4.4 POSITION FIRING 

4.4.1 Necessity of Eye-Shooting 
Methods 

It is unfortunately true that the development of 
weapons and of adequate methods to control those 
weapons rarely keep pace. Control is usually a poor 
second. For example, hand-held guns were installed in 
waist, nose, and tail of heavy bombers, but no me- 


CONFIDENTIAL 




















POSITION FIRING 


51 


chanical provision was made for the difficult problem 
of determining the deflection to take with such guns. 
Under such circumstances an almost intolerable 
burden is placed on a Service Training Command 
which must produce gunners who can estimate ap¬ 
proximately the required deflection by eye. The 
earliest methods of eye shooting were based on a per¬ 
ception of tracking rates. Such methods are properly 
considered in Chapter 5. This section deals with a 
translation into eye-shooting terms of the own-speed 
concept exploited in this chapter. The logic of Sec¬ 
tion 4.1.2 is applicable here. 

4.4.2 Derivation of Rule of Thumb 

Supposing that a bomber is under pursuit curve 
attack, the required defensive deflection is given by 

7 vo . 

sin A = k 2 — sin r. 
v 0 

If it can be agreed that the bomber will operate ap¬ 
proximately at a fixed altitude and a fixed speed vg, 
and if a standard type fighter is expected to attack at 
an approximately known speed, then by means of 
Table 2 a value for k 2 may be selected and sin A will 
depend only on the angle off r. During World War II, 
over Europe conditions standardized very well at: 
Vo = 2,700 fps, v G = 225 mph, altitude = 22,000 ft, 
vt = 325 mph, fighter type G h G 2 . Hence from 
Table 2, 

1004, -'SO + (jD 9.5 + (g£) 7 - 83.8 («), 
and 

100fe = 83+(—) 11 + 8 = 87.4 ((?,). 

VlOO/ V 10,000/ 

A value of 0.85 has commonly been adopted for k 2 
under these conditions. Then 

sin A = 0.104 sin r. 

If a unit of 35 milliradians (called one RAD) is used 
we have values shown in Table 4. If the gunner is 


Table 4. Deflection over Europe. 


t (degrees) 

90 

45 

22.5 

11.25 

0 



135 

157.5 

168.75 

180 

A (milliradians) 

104 

73.5 

39.8 

20.3 

0 

A (RADS) 

2.87 

2.10 

1.14 

0.58 

0 

A (rounded RADS) 

3 

2 

1 

y 2 

0 


provided with a ringsight which is aligned with the 
bore axis and has two (or three) rings subtending one 


and two (and three) RADS at his eye, then under 
the above conditions, if he can estimate the angle 
between the fore and aft axis of his own aircraft and 
the line out to the head-on fighter, he knows from 
Table 4 how much deflection to take. Since the angle 
off is in the plane of action, the amount of deflection 
is independent of the elevation of the plane of action. 
Doctrinally, 179 ’ 183 he looks at the fighter’s position 
relative to the bomber and decides whether the tar¬ 
get is on the 3, 2, 1, or x RAD cone (Figure 5). If 


3 



Figure 5. Key cones for position firing (Army Air 
Forces). 


the angle off is not a key angle, linear interpolation is 
assumed, 61 and, in fact, as the attack develops and 
the target slides from one cone to another the gunner 
is expected to change his deflection continuously. In 
the Royal Air Force (and early Eighth and Ninth 
Air Forces) version of position firing, the gunner held 
a constant deflection of 3 RADS over the zones 60° 
to 90° and 90° to 120°, 2 RADS over the zones 30° to 
60° and 120° to 150°, 1 RAD over the zones 10° to 
30° and 150° to 170°, and 0 RADS over the zones 0° 
to 10° and 170° to 180°. This is called the zone 
system 166> 195 and antedated position firing as de¬ 
scribed above. 

The amount of deflection is now established. It 
remains to consider the line along which this deflec¬ 
tion is laid off. Since the gunner’s eye is in the plane 
of action, which is assumed to remain sensibly fixed 
during the guns bearing phase of the attack, and 
since the vector 0.85v G is assumed to lie in this plane, 
parallel to v G and reversed, it follows as illustrated 
by Figure 6 that except when the target is at an angle 
of A/2 ahead of the beam, the point of aim is always 
on a line connecting the target to the point on the 
gunner’s horizon dead astern. When the attack is in 
the forward hemisphere the target is positioned 
linearly between the pipper of the sight and a point 


CONFIDENTIAL 
















52 


OWN-SPEED SIGHTS 


on the horizon dead ahead. This applies to straight 
and level flight and will be supported by the apparent 
direction of motion of the target as seen by the 
gunner. In certain cases of avoiding action (Section 
3.2.6, form 4) no apparent motion may be evident 
over a time interval of perhaps a second. In this 
instance of a lead plateau, i.e., constant lead, the 
pipper is held between the target and the extended 
fore-and-aft axis of the bomber. 


PLANE OF ACTION 



Figure 6. Line of deflection in position firing. 


4.4.3 Variations in Standard Rules 

The above discussion was predicated upon very 
special operating conditions. It is abundantly evident 
from Section 4.2.4 that changes in bomber speed, 
in operational altitude, and in fighter speed and type, 
require concomitant modification in the eye-shooting 
rules which the gunner is asked to memorize and 
apply. These changes have been made for various 
aircraft and conditions. 32 ’ 62> 63 ’ 152 

4.5 OWN-SPEED SIGHTS 

4 . 5.1 Types 

The mechanical application of the principle of com¬ 
pensating for the speed of the gun mount by intro¬ 
ducing an angle ( k 2 v G sin t)/v 0 between bore axis and 
sight line is as old as military aircraft. The first 
version — a wind vane on the front end of a gun — 
is apparently a British design of 1915 which was im¬ 
mediately exploited by the Germans. 219 Such a sight 
uses the air stream to maintain a vector k 2 v G parallel 
to the flight velocity v G . Modern own-speed sights 
obtain the angle off r by takeoffs from gears fixed in 
the aircraft. The required lead angle is then obtained 
either by a mechanical construction of the vector 
triangle 131 u 0 = v 0 + k 2 v G or by a gear and cam 
calculation of deflection formulas for lead in azimuth 
and elevation. 204 These types will be called the vector 
sight and the algebraic sight respectively. A fourth 


type, a linear correction sight, has been designed for 
the tail stinger of a B-17. 167 Catalogues of the numer¬ 
ous versions of each type are available. 151 - 156 ’ 217 ’ 218 
This section will discuss the Sperry K-13 and K-ll 
sights which are, respectively, examples of the vector 
and algebraic types. 64 

4.5.2 The K-13 Vector Sight 

The inputs to the K-13 vector sight are k 2 v G = v% 
(which is set in on a dial by the gunner), and the 
azimuth and elevation of the bore axis of the gun 
with respect to the aircraft (which are supplied 
automatically to the mechanism by flexible cables 
from the turret gears). The output system consists 
of a collimated reticle image (infinity focus) appear¬ 
ing on a combining glass which rotates about a 
horizontal axis perpendicular to the bore axis to pro¬ 
duce the vertical component of deflection. Preceding 
the combining glass is a mirror which rotates about 
an axis parallel to the bore axis to produce the lateral 
component of deflection as a displacement of the 
reticle image on the combining glass. This optical 
output system is seen in Figure 7. The sight is de¬ 
signed for caliber 0.50 AP M2 ammunition with a 
muzzle velocity of 2,700 fps. The actual mechanism of 
the K-13 is also shown in Figure 7. The points B and 
0 are fixed and the length BO may be taken to be 
proportional to % The length OC varies according 
to the input kv G . The inputs A g (GA) and E g (GE) 
rotate the point C so that OC remains parallel to the 
aircraft’s fore-and-aft axis. The mechanically con¬ 
structed angles S.l{TLD) and A v(TVD) are trans¬ 
lated into the correct rotations of the mirror and 
combining glass of the optical output system. How¬ 
ever, when a mirror rotates through an angle, a 
reflected ray changes its direction by twice that 
angle. This must be accepted and corrected by re¬ 
ducing OB and inserting suitable linkages. A slight 
error arises which has nothing to do with theory or 
manipulation. 

The sight was designed to use k 2 = 0.855. This in¬ 
dicates that theory lagged behind design. No harm 
is done since it is possible to obtain the appropriate 
v G from Figure 3 and use this on the TAS dial set 
for strafing (100 per cent own speed). Similarly, if it 
is necessary to use ammunition with a muzzle ve¬ 
locity different from 2,700 fps, one has only to use 
(2,700 v g )/vq as input. The ratio of OC to OB then 
yields v G /v 0 as it should. 

The K-13 has no range input. Hence any super- 


CONFIDENTIAL 











OWN-SPEED SIGHTS 


53 


COMBINING 
GLASS ^ 



T AS 
KNOB 


I A S 
KNOB 


■ ALTITUDE 
LUBBER 
LINE KNOB 


TRUE AIR SPEED VALUES 
READ ON THIS DIAL ARE 
SET INTO SIGHT ON SIDE 


NOTE! THIS SCHEMATIC SHOWS SIGHT WITH 
GE AT +1600 MILS, GA AT 0 MILS T A 
S s 300 M PH ARROWS SHOW: UP 
ELEVATION, RIGHT AZIMUTH, UP TVD, 
RIGHT TLD, INCREASING T A S 


Figure 7. K-13 vector sight schematic. (Courtesy of Sperry Gyroscope Company.) 


CONFIDENTIAL 

















54 


OWN-SPEED SIGHTS 


elevation allowance for gravity drop can depend only 
on direction of fire. The design superelevation used 
is 6 cos E g milliradians, with a slight dependence on 
azimuth. (The dependence is a minimum on the nose, 
since future ranges will be a minimum, and a maxi¬ 
mum on the tail, with a spread of 2 milliradians for 
E g = 0.) Taking, roughly, gravity drop to be 5 1 2 yd, 
and a mean velocity v = 2,500 fps, it follows that the 
allowance of 6 milliradians corresponds to a range of 
approximately 800 yd. If it is decided that this is too 
great, it may be reduced by setting the elevation dial 
appropriately during alignment of gun and sight. 

The nature of the K-13 is such that there is no 
restriction on the azimuths it will accept and there is 
but minor restriction on elevations (+85° to —85°). 
It is suitable for use at any gun position. 


INPUT DIALS FLEXIBLE SHAFT INPUTS 
IAS ALTITUDE LEGEND 



Figure 8. K-ll algebraic sight schematic. (Courtesy 
of Sperry Gyroscope Company.) 


4.5.3 The K-ll Algebraic Sight 64 

The inputs to the K-ll algebraic sight differ from 
those for the K-13 in that IAS and altitude are to be 
inserted instead of k 2 v G . The optical output system is 
the same. 

It is an easy exercise in spherical trigonometry to 
show that, with good approximation, 

^ Z sin A g 

L 1 + Z cos A g cos E g 

Z cos A g sin Eg 

Av — - r/ - - ’ 

1 — Z cos A g cos E g 


where 



Vo 


Instead of physically constructing angles A L and Av 
as does the K-13, the K-ll mechanically works 
through these formulas as indicated in Figure 8. 

In such circuits (1) differentials add algebraically 
two input rotations, (2) one-dimensional cams give a 
function of a single variable, (3) two-dimensional 
cams give a function of two variables, and (4) a rack 
and pinion translates a rotation into a displacement. 
It follows that multiplication must be effected by 
logarithms. Hence, in particular, a log cosine cam is 
restricted in size since log cos 90° is negatively in¬ 
finite. Consequently, algebraic sights can only be used 
in nose and tail positions or, more generally, in some 
restricted region, the K-ll being restricted to a cone 
of radius 60° about the nose. 

The original design of the K-ll called for inputs 
of altitude and IAS. It computes TAS. The design 
k 2 is given in the following Table 5. 


Table 5. Design k 2 of K-ll algebraic sight. 


Altitude (feet) 

k 2 (AP M2) 

k 2 (API M8) 

0 

0.805 

0.856 

6,000 

0.835 

0.888 

12,000 

0.859 

0.913 

18,000 

0.873 

0.928 

24,000 

0.879 

0.934 

30,000 

0.877 

0.932 

36,000 

0.864 

0.918 


It is again evident that design preceded theory. 
If it is desired to use the optimum v% from Figure 3 
one proceeds as follows. Let a be relative air density 
(NACA). Then the v G {TAS) used by the K-ll is k 2 • 
[/AS] • (j~'% where k 2 is that of Table 4. But one 
wants to set on the IAS dial Vg- Hence fc 2 [/A$] • 
= [/AS] and it follows that the altitude dial 
may be set and left at the altitude corresponding to 
the solution a of the equation = k 2 (a). For AP M2 
ammunition, this altitude is 10,450 ft and for API 
M8 ammunition it is 7,400 ft. (In the actual sight 
IAS is set opposite altitude. Hence vq is set opposite 
these values of altitude.) A second method is to put 
the sight at “strafe,” which is 100 per cent own 
speed, and set Vq opposite zero altitude. With the 
sight in hand it is easily seen that the two schemes 
are identical. 


CONFIDENTIAL 



































SUPPORT FIRE 


55 


4.5.4 The Class B Errors of Own- 
Speed Sight 

The errors in gun pointing committed by a gun- 
sight are due to (1) errors in the inputs, (2) mechani¬ 
cal errors, such as back lash, caused by the construc¬ 
tion, (3) errors attributed to engineering compromises 
in the design, (4) errors in the gunner’s manipulation, 
and (5) errors caused by a design based on inaccurate 
formulas. 

Input errors arise by using an incorrect v G or v Q 
(over the life of a barrel, v 0 may drop 200 fps). Any 
percentage error in v G or v 0 appears as the same per¬ 
centage error in lead as differentiation of A = ( v G 
sin t)/vq shows. Input errors in A G and E G may also 
occur because the gun was incorrectly boresighted, 
because the aircraft has an angle of attack, 131a or 
because the aircraft is moving forward crabwise. 
Since A G and E G are picked off relative to the aircraft 
and not relative to the true velocity vector, the guns 
may not lie in the correct plane of action. 

Errors of types (2) and (3) are analyzed by labora¬ 
tory bench tests. 6 The results are not described here. 
Type (4) errors are dealt with in airborne experi¬ 
mental programs. (See Chapter 7.) This section con¬ 
siders only errors of the theoretical design of type (5). 

These theoretical faults are called Class B errors. 
Suppose that a sight is perfectly constructed accord¬ 
ing to the blueprints, that it is operated perfectly, 
and that there is no dispersion. The bias in gun point¬ 
ing that still exists is caused by the systematic failure 
of the sight in computation. When the own-speed sight 
is used against a pursuit curve, the choice of k 2 = 
0.85 or even k 2 oP t must lead to Class B errors in the 
plane of action, since the factor chosen has only an 
average value. There is also an error normal to the 
plane of action because the path of the fighter sags 
below the plane of action of the instant of fire and 
because the gravity drop allowance is not a function 
of range. 

It is desirable to express these errors as a function 
of range and angle off 21a rather than as a function 
of time along particular courses. Closed deflection 
formulas such as equation (19) in Chapter 2 make 
this possible. As an example of this point-function 
technique, consider the family of all aerodynamic 
lead pursuit curves, with specified v B , vf, and alti¬ 
tude', which lie in a horizontal plane of action. The 
correct lead for any r and r may be computed by 

e Such tests were made at Northwestern University under 
Contract OEMsr-1276 as Project No. 14. 


equation (19) in Chapter 2 with the approximate 
theory of Section 4.2.3. Then, for an own-speed sight 
set at k 2 = 0.85 and at k 2 op t = 0.77, the Class B error 
in the plane of action l4a can be presented cellularly 
as in Figure 9. 

ERROR IN PLANE OF ACTION IN MILLIRADIANS 
+ VALUE MEANS OVERLEAD 


v 6 = 225 MPH v p =315 MPH H = 9500 FT 



Figure 9. Typical Class B errors of an own-speed 
sight. 


By superposition of any relative pursuit curve on 
Figure 9, the way in which the error varies along that 
curve may be inferred. Characteristically, the sight 
overleads initially, leads properly at midranges, and 
then underleads. When a different percentage is used, 
the range at which the lead is correct changes. But 
the bullet pattern still walks slowly through the tar¬ 
get. This is obviously preferable to a fixed bias in 
pointing since the dispersion pattern rapidly becomes 
dilute as the target moves from the MPI. Figure 9 
shows that the range at which the pattern is centered 
is tactically better for k 2 oP t = 0.77 than for k 2 = 
0.85. (For this example the lead required by sag, 
normal to and downward from the plane of action, 
varies from 1 to 5 milliradians. It is possible to 
combine these values with the required allowances 
due to gravity and windage jump and assess Class B 
performance normal to the plane of action. This will 
not be done here.) 

4.6 SUPPORT FIRE 

4.6.1 Support Fire Situations 

A presumable implication of a tightly massed 
bomber formation is that cross or support fire is 
quite possible. The guns of one bomber can be 


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56 


OWN-SPEED SIGHTS 


brought to bear on a fighter attacking another 
bomber of the group. Massed firepower f has, in fact, 
been viewed with dejection by opposing air forces. 174 

If the specialized own-speed sight is to be used in 
support fire the pipper can no longer be held on the 
target and the sight must be used in combination 
with some eye-shooting rule. During the closing 
months of World War II, over Europe there was a 
need for such application. The German Air Force 
massed fighters astern of a bomber formation and 
rode up the tails of all bombers almost simul¬ 
taneously. This was the Company Front attack. 65,175 
The nature of the tactic masked out upper and lower 
turrets, leaving tail and waist positions to counter 
this saturation attack. Other support fire situations 
arise when a neighboring bomber is under pursuit 
attack or under direct frontal attack. Assuming that 
waist and tail positions are equipped with own-speed 
sights (and not even a majority were) that sight 
must be used for a job that completely violates its 
design. There was no alternative since fire control 
capable of meeting general attack paths was not 
available for nonturreted emplacements. 

4.6.2 Use of Own-Speed Sights 

Suppose for the moment that the speed input to 
the own-speed sight is full TAS of the gun mount, 
i.e., k 2 = 1. Then the bullet will go along the gunner’s 
line of sight in the air mass with an initial velocity of 
u 0 . Hence the gunner must displace his point of aim 
to allow for the crossing speed vt sin a of the target. 
The deflection he must take, by eye, is (vt sin a)/u 
radians. Even this does not take into account the 
correction required by the curvature of the target’s 
path. 160 However, it has been suggested that for many 
important support fire situations the target will be 
on a near level with the supporting bomber and will 
fly sensibly in a straight line. 43,66 For this case, if the 
target speed and the average bullet speed may be 
fixed approximately, the gunner’s problem is reduced 
to the estimation of the approach angle a. The 
amount of his deflection is then given by some such 
memorized sequence as 157 

a (degrees): 0 10 20 30 45 

Lead (RADS): 0 1 2 3 4 

This deflection is laid off along an extension of the 

f But it is quite essential that a distinction be made between 
a plot of the number of guns that can be brought to bear on 
each point of a sphere surrounding the bomber unit, and a 
plot of the accuracy in deflection with which supporting guns 
will point at positions on that sphere. 


target’s fuselage (not in the direction of the target’s 
apparent motion). Ingenious methods of estimating 
a have been given, which note the relative position 
of an empennage as seen superposed on wings. 43 The 
ability of a gunner to recognize support fire situ¬ 
ations 155 and to estimate approach angles 153 have 
been studied by controlled experiments with groups 
of gunners. 

Much of the literature on support fire 30, 162,163 is 
concerned with variations in rules such as those 
quoted above and with pro and con arguments about 
shifting the own-speed sight, when used in support, 
from k 2 opt to k 2 = 1 . The subject is probably a dead 
end as far as future developments are concerned. It 
will not be pursued further here. 

4.7 SUMMARY 

Section 4.1 introduces own-speed sights as devices 
designed to counter the special case of a pursuit curve 
attack on a defending bomber. It discusses the argu¬ 
ments for and against such equipment. 

Section 4.2 shows that in shooting against a pure 
pursuit curve a large fraction of full own-speed allow¬ 
ance is made since the target path curves forward 
but slightly during the bullet’s time of flight. This 
section continues with a discussion of the factors 
affecting deflection to be taken against an aero¬ 
dynamic lead pursuit curve, gives optimum rules sum¬ 
marizing detailed calculations, and reduces the prob¬ 
lem to simple charts which could be used in the air. 

Section 4.3 discusses the validation of the theory 
of the previous section in three directions (1) the 
demonstration that curves approximating pursuit 
curves were flown extensively in combat, (2) the 
agreement between the predictions of theory and the 
deflections demanded by paths actually flown by 
fighters during gunsight assessment programs, and 
(3) the agreement between the predictions of the 
crude theory and the exact analytic theory of 
Section 3.4.2. 

Section 4.4 considers position firing, with the aid 
of which the gunner is to estimate his deflection by 
eye, i.e., he is his own compensating sight. 

Section 4.5 discusses briefly the mechanical fea¬ 
tures of the vector and algebraic types of own-speed 
sight, and partially reviews Class B errors. 

Section 4.6 is concerned with support fire with an 
own-speed sight in which eye-shooting rules are 
combined with the automatic compensation for own 
speed. 


CONFIDENTIAL 




Chapter 5 


LEAD COMPUTING SIGHTS 


5.1 INTRODUCTION 

5.1.1 First-Order Nature of Sights 

F ire control should be flexible enough to supply 
accurately, quickly, and continuously the aim¬ 
ing allowance required by a target traversing an ar¬ 
bitrary path relative to the gun mount. Unlike the 
own-speed sights of the preceding chapter, the 
mechanisms considered below do not assume that 
the target’s approach angle is behaving in the highly 
special way required by an aerodynamic lead pursuit 
curve. In this sense, lead computing sights can handle 
arbitrary target paths. They are, however, inflexible 
in another and important sense. Basically, they assume 
that the target’s track relative to the gun mount is a 
straight line (sensibly straight over the time of flight 
of the bullet). Operation is founded on the theory of 
Sections 2.2.4 and 2.2.5. Despite the calibration 
concept (Section 5.3.5), which presumably permits 
the sight to deal with a class of curved paths, these 
first-order mechanisms do not solve the general fire 
control problem adequately, since accuracy against 
one class of target paths is gained at the expense of 
lowered performance against some other class. Be¬ 
sides describing those lead computing sights used in 
inhabited turrets during World War II, it is a re¬ 
sponsibility of this chapter to enlarge upon and 
justify the statements made above. Somewhat more 
generally, the chapter is to relate lead computing 
mechanisms to the points of the initial sentence of 
this introduction. 

5.1.2 Rate Deflection Formulas 
Mechanized 

The most important deflection formulas of Chapter 
2 were those based on the angular rate of the gun- 
target line. The reason is that the introduction of 
angular rate deletes the target’s approach angle 
which is not measured. In simplest form the lead was 


the difference between a kinematic deflection and a 
ballistic deflection. The latter deflection depended 
mostly on gun position and range so that one would 
expect it to be mechanized, conceptually at least, 
in a fashion independent of the kinematic deflection. 
Consequently, we must consider as a basic computing 
device a machine (1) which will convert range (and 
altitude and perhaps other variables) obtained stadi- 
ametrically or by radar into a time-of-flight multi¬ 
plier t m , (2) which will obtain the present angular 
rate co of the gun-target line by a gyroscope, variable 
speed drive, or a tachometer, and (3) which will 
combine these to yield the kinematic deflection t m o). 

The curvature correction factor h of equation 
(12) of Chapter 2 depends not only on future range, 
as does t m , but also on the direction and amount of 
curvature. The machines considered in this chapter 
can take curvature into account only by assuming 
some particular average behavior on the part of the 
target. In particular, information on rate of change of 
range is neither given to nor accepted by these de¬ 
vices. The attitude is adopted that h and t m , and even 
trail, are at our disposal and are to be chosen to give 
optimum performance over a special class of target 
paths. However, this procedure is not nearly as re¬ 
strictive here as was its equivalent in the case of the 
own-speed sight. (Suppose for example that the cur¬ 
vature correction appropriate to a pure pursuit curve 
is adopted. Then, even if the target is not flying a 
perfect guns bearing attack, we would still expect to 
have a good approximation to its actual curvature. 
The same remarks apply to support fire. The topic is 
investigated in more detail below.) 

In obtaining the angular rate co it is important to 
distinguish between the actual angular rate of the 
gun-target line in the air mass and the tracking rate 
as determined at the gun position. For example, if 
gun mount and target are chasing each other in a 
circle of radius R at equal speeds, co = vg/R • The 
tracking rate, on the other hand, is zero. (For this 
reason, kinematic deflection is a somewhat inaccurate 


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57 


58 


LEAD COMPUTING SIGHTS 


term.) In general, any mechanism which measures 
angular rates with respect to the gun mount will err 
whenever the axes of that mount are in curved flight. 

5.1.3 The Disturbed Reticle 
Principle 

The sights considered in this chapter are of the 
disturbed reticle a type. The gunner controls the posi¬ 
tion of the bore axis of his gun either manually or by 
motor control. The computing unit lays out the 
deflection between the bore axis and the line of sight. 
The gunner has, then, only indirect control over the 
line of sight which he is required to keep on target. 
In general, a given motion of the gun results in a 
different motion of the line of sight. One disturbs the 
line of sight instead of controlling it. 


5.2 APPARENT MOTION EYE SHOOTING 

5.2.1 Historical Reason 

In the preceding chapter it proved possible to 
supply rules to a gunner provided only with a fixed 
sight by which he became his own compensating 
sight. Similarly, it is possible to give rules by which 
the gunner becomes his own rate-time sight. This 
procedure was in effect in our air forces in the early 
■days of World War II. It preceded position firing 
and all gunsights with the exception of the Sperry 
K-3 installed in the Sperry upper turret of the B-17. 
(Actually, the K-3 itself was only a pilot model and 
not really intended for extended use. 67 ) Although the 
digression of this section is primarily of historical 
interest, it demonstrates what must be done when 
design of a control mechanism lags the weapon 
installation and is not coordinated with that installa¬ 
tion. In some form or other it is not unlikely that 
such situations will always arise during a war. Eye¬ 
shooting methods are of dubious efficiency in the air. 
An inordinate amount of training is needed to achieve 
any results at all. A close analogy is flying by the seat 
of the pants in weather of zero visibility. 

a Modern gunsights whether computing, or own speed, or 
fixed, use an optical system in which a reticle pattern scratched 
on a plate located in the focal plane of a collimating lens 
system appears as an image on a combining glass mounted 
at an acute angle to the line of sight. To the observer, the 
reticle appears to be at infinity, i.e., on the target, and the 
user has no problem in close-far distance adaptation nor need 
he hold a fixed eye to sight distance to preserve the angular 
dimensions of the reticle. The eye may be moved about over 
the field. 


5.2.2 The “Elephant” Method 


An early method of the type referred to above 
was the USN Elephant method. 184 ’ 203 The gunner 
(1) tracks with pipper on target until the range is 
650 yd (stadiametric estimation), (2) stops pipper 
and observes how far the target moves across the 
sight framework in two-thirds of a second (while the 
word elephant is said aloud), and (3) positions target 
an equal and opposite distance on the other side of 
the pipper and starts firing. The two-thirds of a 
second deflection is decreased in proportion to the 
ratio of actual range to 600 yd. This procedure may 
be considered analytically. If the target is on a pure 
pursuit curve the required deflection is 



1 Vt 

2 Vq 



T) 


by Section 4.2.1. The angular rate of the gun-target 
line is 

v G sin r 

co =- > 

r 


where r is the range. If co does not change radically 
the gun is held still for 



seconds, where t m is a time-of-flight multiplier. 44 If 
r = 1,800 ft and v 0 = 2,700 fps, then t m is approxi¬ 
mately two-thirds of a second. The factor k 0 varies 68 
from 0.85 to 0.95 with 0.90 as an acceptable mean. 
Suppose now that the gunner makes an error dt in 
estimating time. Then, assuming that he can de¬ 
termine and lay off the resulting motion perfectly, 
his error is 

(1^600 + *)“ “ tmU = *’< jsin7 (°- 112 + 1000 ^ 


milliradians, where r is in yards and v G is in yards 
per second. If for all gunners the average dt is zero, 
the average shooting is biased by about 12 sin r milli¬ 
radians, for a bomber at 225 mph, because of the 
neglect of the curvature correction in the rules. If 
the rules were adjusted to allow for this, and if, then, 
the average absolute error in estimating time were 
1^1 = 0.15 sec > at GOO yd, the average absolute 
error in lead would be 27.5 sin r milliradians. 

It was appreciated 184 that a correction must be 
made for approach angles that are not sensibly zero. 
The gunner is to use some point on the extended 
fuselage of the target from which to lay off his de- 


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BASIC THEORY OF LEAD COMPUTING SIGHTS 


59 


flection as arrived at above. This is obviously an 
allowance for ballistic deflection. It is doubtful that 
a combat gunner could do this. Further discussion is 
possible, 1 but in view of the real practical difficulties 
in carrying out the process in the air (jumping sight, 
etc.) the subject will not be elaborated. 


5.2.3 The “ABC” Method 


The USAAF used a method quite similar to the 
Elephant method. This was the ABC method, in 
which ABC , when said aloud, was presumed to have 
a duration of three-quarters of a second. The discrep¬ 
ancy in duration of time in the two methods is ex¬ 
plained by noting that general firing conditions, as 
opposed to pursuit curve attacks, were in mind. 
The bullet was to have an average speed of 2,400 
fps over 1,800 ft. It is evident that this errs in exactly 
the wrong direction in countering pursuit curves. 
Against a target on a parallel course at the same 
speed it gives no lead (like the Elephant method) 
and so does not even allow for trail. The method, in 
fact, will give the correct lead when the target is 
crossing at right angles to the bomber’s track at a 
range for which v = 2,400 fps. 1 For this case 

vt . vt . r r 3 

A = q — sin a, oi = — sin a, t m = q — = - = - • 

v 0 r Vo v 4 


For a target on a rectilinear course the error of the 


method when used perfectly is co 
where 44 

Vg sin r — Vt sin a 


V 4 600 / 


r 


r 

tm 

Vo 


Vg sin r 
Vg sin a 


- 1 


(See Sections 2.2.1 and 2.2.3.) After computation, it 
is seen that the errors are really serious. 


5.2.4 The “Apparent Speed” Method 

The two preceding methods fixed a time interval 
and determined how far the target moved in the 
frame of the sight. It is possible to fix the distance to 
be moved, e.g., the ringsight radius, and determine 
how long this motion takes. This is the Apparent 
Speed method (USN). A ringsight whose angular 
radius is 35 milliradians gives the deflection, without 
trail, to take against a target with a speed advantage 


of 50 knots, abeam, and on a parallel overtaking 
course at a range for which the average speed of the 
bullet is 2,413 fps. (Formula (1) of Chapter 2 with 
Vo/q = 2,413 gives A = 35 + 0.045^ milliradians, 
where v G is in feet per second.) Now, for example, if 
a target at 2,000 ft takes 0.5 sec to cross the radius, 
its crossing speed is (35)(2)/(0.5)(1.69) = 83 knots 
(a memorized figure of 80 is used) and the deflection 
laid off is 80/50 = 1% RADS (1 RAD = 35 milli¬ 
radians). This method is quite equivalent to the 
ABC method. 1 

5.2.5 Critique 

The rules of thumb discussed in this section are 
ingenious but unsatisfactory on both theoretical and 
practical grounds. Theoretically they neglect trail 
and curvature effects and will not function when the 
gun mount is on a curved course. Moreover, it is 
assumed that the angular rate observed will persist 
while deflection is laid off. Ballistics are not permitted 
to vary with range and altitude. On the practical 
side they are difficult to apply at any except key 
ranges, even assuming the target would drift in a 
smooth path across an airborne sight. 

5.3 BASIC THEORY OF LEAD COM¬ 
PUTING SIGHTS 

5.3.1 Range and Tracking Rates 
and Smoothing 2 

Since ballistic effects usually constitute a minor 
part of the total deflection, and since these effects 
may be presented quite accurately by subsidiary 
mechanisms, let us consider generically the nature of 
a lead computing sight designed to produce a kine¬ 
matic deflection which is the product of a time-of- 
flight multiplier, t m , and the present angular rate of 
rotation, co, of the line connecting gun to target. 

In order for a sight to obtain t m it must be supplied 
with some sort of continuous estimate of the present 
range to the target. The range is usually obtained 
stadiametrically by the operator. This means that 
an optically presented reticle image of some pattern 
(parallel bars, dots in a circle, a ring) can be changed 
in size by the gunner. The remaining two elements of 
the required simple proportion are the actual size of 
the target and the size of the target in the scale of 
the instrument. The operator must set in the latter 
small instrumental dimension before the engagement. 


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60 


LEAD COMPUTING SIGHTS 


The changes of the reticle required to keep the tar¬ 
get framed, when compared with the fixed input 
size, permit the sight to obtain the range. The range 
p supplied in this fashion by the operator will not, in 
general, be the correct present range r. The differ¬ 
ence p — r is the ranging error and is a function of 
time. Two ideas immediately occur. Why not smooth 
the range error p — r before using it? (This is in 
analogy with communications engineering practice 
in Avhich irregular and oscillating terms are called 
noise and are subjected to smoothing circuits.) 
Smoothing of the range is not done in the devices con¬ 
sidered in this chapter. And why not use the rate of 
change of range p to estimate future range? This 
estimate would be made by p + t m p, just as future 
position is estimated by t m u. In practice, p itself is 
poor and jumpy and p is not used. With radar input 
of range both of these ideas become more attractive. 

In order for the sight to obtain co it must be sup¬ 
plied continuously with the angular position r of 
the target. Given the position continuously, the sight 
can obtain the required rate f = co. The gunner must 
keep the center of the reticle image (the pipper) on 
target. He will generally err in this and give to his 
sight a, which is the angular position of the pipper, 
and is irregular and oscillatory. The difference r — a 
is called the tracking error or noise. For angular po¬ 
sition, both of the points of the previous paragraph 
are carried out. This datum is smoothed and the 
angular rate is computed, as, of course, it must be. 

The simultaneous requirements of ranging and 
tracking (and triggering) are mutually inhibitory. 
Performance on either one improves markedly if the 
other is missing. In general, tracking is better than 
ranging in the sense that the error is smaller and more 
rapidly random. Thus smoothing on the tracking in 
angle is feasible. 

In this chapter we shall assume that the gun mount 
is in uniform motion. This is equivalent to saying 
that the tracking rate r is equal to the true angular 
rate of the gun-target line co. Then those sights 
which measure rates relative to the gun mount and 
those that measure it relative to the air mass may be 
subsumed under the same theory. 

5.3.2 The Exponential Smoothing 
Circuit 

The most common smoothing system is the ex¬ 
ponential smoothing circuit which can be realized in 
fire control electrically, mechanically, and gyro- 


scopically. In fact, this is the only type of circuit 
exploited in the devices of this chapter. In Figure I 
the basic angles of the discussion are illustrated. (To 
make the argument transparent, it is assumed that 
the motions of gun mount and target are coplanar. 


B 



Figure 1 . Basic angles of lead computing sight 
theory. 


The notation X instead of A k will be used throughout 
this chapter for kinematic deflection.) If the tracking 
were perfect, a = r and \ = y — t, so that X = 
y — t. The lead produced should be X = t m r . Hence, 
the sight equation would be 

tmX + X = t m y. 

As will be seen below, the equation actually mecha¬ 
nized by lead computing sights is not the preceding 
one but is 

(1 + a)t m \ + X = t m y • 

The constant a leads to smoothing and is a conse¬ 
quence of the mechanization adopted. From this 
equation one obtains 

at m \ -f- X = t m <j , (1) 

where X is now the actual lead produced by the sight 
since one takes g = y — X to obtain equation (1). 
This is the basic equation of the theory. 

5.3.3 Physical Realization of the 
Sight Parameter 

The constant a is called the sight parameter. It can 
be given a physical interpretation. From equation 
(1), it follows that 

X — t m (G (zX). 

If a fourth line is added to Figure 1, making an angle 
a — a\ with the reference line, it follows that the 


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BASIC THEORY OF LEAD COMPUTING SIGHTS 


61 


sight is computing lead as if the target were on this 
fourth line. Describing this line by GY , it is evident 
that as the system rotates about G, the ratio of the 
angle YGS to the angle SGB remains constant at a. 
As the deflection changes, the proportionate posi¬ 
tion of GB, GS, and GY does not change. In gyro¬ 
scopic sights, the line GY has a definite existence. 
It is the spin axis of the gyro. 


5.3.4 Discussion of the Circuit 


A discussion of equation (1) is necessary and inter¬ 
esting. 

Weighted Averages 

We can first see in what sense smoothing or aver¬ 
aging is effected. The time-of-flight multiplier is a 
function of time. But the change to a new variable z 
by dt = t m dz will eliminate t m , and an equation with 
constant coefficients arises. It is 


d\ da 

a Jz + X= dz 


The integrating factor is e z,a , and multiplication by 
this and integration by parts give 


A — XqC Z/fa -\- 


r z/a r 

— 1) Jo 


where z 


a(e z/a - 1) 
r l dt 

= I — , 2(0) = 0, and X(0) 
JO t m 


dz 


( 2 ) 

Ao, with time 


originating at t = 0. The integral in equation (2) 
suggests the exponentially weighted average of da/dz. 
The total weight is 

w(z) = fe z,a dz = a(e z/a - 1) . 

Jo 

If we use (da/dz) av to mean the average of da/dz 
weighted in this sense, then equation (2) can be 
written 2 

The entire second term of equation (3) may be called 
the transient, since it becomes small as z increases. 
The weight function e z/a is such that decreasing sig¬ 
nificance is attached to information further and 
further back in the past. If a is large, e z,a falls off 
slowly, and the smoothed present value depends 
more heavily on the past values. 

Damping of Oscillatory Tracking 
From a somewhat different point of view the 
smoothing effect of the circuit can be clarified by a 


simple example. Suppose that the target is moving 
in a circle around the gun at uniform angular speed co. 
Then t m is constant. Suppose that the tracking rate 
oscillates around the correct value according to 
a = co + K sin nt. Then the steady state solution of 
equation (1) may be written 

Kt 

X = t m o) + . m - sin (nt — tan -1 ant m ). 

VI + avtl 

If a were zero the lead produced would be 
X = tm co + Ktm sin nt. 

The ratio of the lead error amplitude with a nonzero 
a to the amplitude with zero a is 

1 

Vl + a 2 » 2 & 

This effect is called damping. 


Decay of False Leads and Slewing Routines 
The initial lead of the sight X 0 may well be false, 
since under slewing to get on target the sight feels 
that a fast target is at hand. It is important to know 
how rapidly such false leads decay if the rapidity of 
lead computation is to be assessed. Suppose the sight 
axis is displaced from the bore axis by an amount X 0 . 
If the gun is not moving, 7 = 0. Hence by the equa¬ 
tion connecting lead to gun position, we have 

(1 + a)t m \ + X = 0 _ t 
and X = A 0 e (1+a) ' m * 

The response of the circuit to a particular form of gun 



0 12 3 

t 


Figure 2. Drift of sight line back to bore axis. 

motion is shown in Figure 2. It follows from the 
above equation that X decays to 1/e of its original 
value in (1 + o,)t m sec. This time is called the time 


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62 


LEAD COMPUTING SIGHTS 


constant of the sight mechanism. (If t m = 1 sec, the 
resulting time constant 1 + a is sometimes called 
the sight response factor. 205 ) Since t m decreases with 
decreasing range, in slewing the gunner is frequently 
instructed to hold the range setting at a minimum 
until he is on target with the sight pipper. 

In an attempt to minimize the time required for 
the transient (or false lead) to decay, slewing routines 
other than the simple one mentioned above have been 
proposed. The problem is approached analytically. 120 
Suppose that a target has a constant angular velocity 
r 0 . The true lead is then taken to be t m T 0 . Let the gun 
position be, originally, t 0 , behind the target. Sup¬ 
posing that the maximum slewing rate of the turret 
is A radians per second, that gun motion is required 
which minimizes 69 the time interval in which the gun 
reaches and holds its correct position (7 = r 0 + tr 0 + 
t m To) and in which the sight line also attains its cor¬ 
rect position (<7 = r 0 + tr Q ). The solution consists in 
slewing at the maximum rate until the gun is beyond 
its true position at a time h, and then in slewing back, 
again at the maximum rate —A, until it reaches the 
true position at a time U. The latter time 1 2 is readily 
picked out by the gunner as the time after ti when 
the sight is on target. The real difficulty lies in telling 
the gunner how to determine h, i.e., how far to go 
past the target. This question has not been answered 
and this slewing routine has not become standard 
operational procedure. 

Amplification of Tracking Errors 

If attention is turned to the relation between 
tracking errors and gun errors, it will be seen that 
errors in the tracking are amplified when translated 
into errors in gun pointing. Consider the equation 

(1 + a)t m cx + a = at m 7 -f 7, (4) 

which arises from equation (1) if we put X = 7 — a. 
Suppose that the sight line is oscillating according 
to o- = A sin nt. Then, for constant t m , the steady- 
state solution of the gunsight equation is 

7 = MA sin {nt — <fi), 

where 

m-[/IEl±W ( „ 

' 1 + a 2 t* m n 2 

and 0 is a phase angle whose small value need not be 
considered. M may be interpreted as the ratio of the 
amplitude of gun oscillation to the amplitude of the 
sight-line oscillation. It is called the amplification 
factor and is plotted for typical values in Figure 3. 
Amplification deceives the gunner, since he may con¬ 


sider his tracking good (amplitude of error 5 milli- 
radians) and yet his guns may be sweeping over the 
target with an amplitude threefold increased. It is 
evident that as a approaches zero the amplification 
increases, and as the frequency n/ 2t increases the 
amplification increases but tapers off to the value 
(1 -f- a)/a. This is a good reason why sights use 
circuits with nonzero parameters. 



Figure 3. The amplification of oscillatory tracking 
errors. 


In actual tracking, errors are committed not only 
along the path of the target but also at right angles 
to that path. It is natural to ask if the two com¬ 
ponents of noise can be treated independently insofar 
as amplification is concerned. The answer is yes. 70 
(There is a very slight cross effect on the vertical 
amplification factor which reduces it by about 0.1 
per cent.) For this two-dimensional case, however, 
it may be pointed out that for certain combinations of 
frequency and relative phase of the componential 
tracking noises, the resulting amplified gun motion 
may be such that the gun never gets on target, even 
though each component of the tracking crosses the 
target position. (Obviously, if a L = A sin nt and 
(xv = A cos nt are the lateral and vertical tracking 
noises, the gun describes a circle of radius MA.) 
Tracking errors are in general not so regularized 
that this is matter for concern. 

Operational Stability 

The question of operational stability may be clari¬ 
fied through equation (4). Suppose initially that 


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BASIC THEORY OF LEAD COMPUTING SIGHTS 


63 


o- = Y = 0. If the gun is given a sudden jerk, the 
response of the sight line is determined by 

a 

o-o = —— To , 

1 + a 

where t 0 is the initial jerk. Then (1) if a is positive, 
the initial sight response agrees in sense with the gun 
motion and the sight is called operationally stable, 
(2) if a is zero, the sight does not respond immediately, 
there is no impulsive effect and the sight seems slug¬ 
gish, and (3) if a is negative, the sight is a fast starter, 
but in the opposite direction to the gun, and is called 
operationally unstable. In the third case the gunner 
is deceived since a small corrective jerk in what 
should be the right direction causes the sight to move 
initially in the opposite direction. Such sights can 
be built, however, and will function in the sense that 
if the guns are started and continue tracking, the 
reticle will ultimately start back in the right direction, 
overtake the target, and get into lead position. 
Nevertheless it does not seem advisable to use nega¬ 
tive values of the sight parameter because of the 
disturbing effect on the gunner, and also because 
the weighting (see Weighted Averages”) would 
attribute more importance to old values than to the 
new ones. 

Delay in Lead Computation 

The considerations of the previous paragraphs were 
based on the interpretation of equation (1) as a 
smoothing or damping circuit. Such systems may 
also be thought of as exponential delay circuits in the 
sense discussed below. If equation (1) is differenti¬ 
ated and if the second derivative of X is neglected, 
one obtains the expression 

X tm,G at m , ( t m O’) j (6) 

di 

as an approximation to X. Admitting that tracking 
is correct, equation (6) shows that the circuit errs in 
producing the correct deflection t m a (unless t m and fl¬ 
are constant in time). Denoting the correct lead by 
X 0 , equation (6) is rewritten in the form 

A — Ao — at m . 

dt 

X has approximately the value that X 0 , the correct 
lead, had at a time at m in the past. (Put dt = at m .) 
In this sense the sight’s answer is always stale and 
becomes staler as a is increased. The delay must be 
accepted if smoothing is desired. 

From one point of view, the delay is desirable. 


Suppose that instead of t m we use the time of flight 
over present range t p . Then, referring to Section 
2.2.5, one recalls that on the! incoming leg of a target 
course t p oo is too large a kinematic deflection. If a 
delay circuit is qsed, the correction it makes to 
X 0 = t p o) is in the right direction, if a is positive , since 
on the incoming leg lead is increasing. A similar 
compensatory effect is exercised on the outgoing leg. 
In spite of this effect, it is now generally conceded 
that a sight should be calibrated, i.e., that t m should 
be chosen to give best results over a given set of 
tactical circumstances. Hence this behavior cannot 
be construed as favoring only sights for which a > 0. 
For any value of a, calibration can allow for delay. 
(See Section 5.3.5.) 

The exact solution of the type equation (6) is 
needed if delay is to be determined precisely. For 
equation (1), this exact solution is 9 

X t m <j at m ( tm o’) 

dt 

+ e z,a |\o — t m ooo + —(^mo-)oj + F(z), (7) 

where z = | dt/t m , and E(z) is the error term. 

Jo 

Choice of Sight Parameter 

From the discussion of this section, it is evident 
that there is no rational way in which an optimum 
value for a can be selected. There are factors which 
want a small and others which would prefer it to be 
large. If we first agree to use a positive value for a 
in order to have operational stability and proper 
weighting of the newest data, then the factors in¬ 
volved can be summarized as follows. For large a, 
smoothing and damping are better, but, on the other 
hand, perhaps too much weight is assigned to past 
information. As a increases, transients (false leads) 
take longer to decay, and the delay in putting out 
correct deflection is greater. These are not desirable 
features. On the other hand, as a increases, the 
amplification of gun motion is decreased which is 
probably desirable. An engineering compromise 
must be made. The sights of this chapter use 
parameters with values somewhere in the range 0.2 
to 0.5. 

Since t m has an effect paralleling that of a (a and 
t m occur as a product in connection with the above 
factors), it is reasonable to suggest that one should 
have a vary with range to effect compensations. This 
situation will be dealt with briefly in Chapter 8. 
But it is worth emphasizing now that operational 


CONFIDENTIAL 



64 


LEAD COMPUTING SIGHTS 


errors of simple manual ranging and tracking usually 
overpower such refinements. It is another case in 
point of improving inputs before refining the blind 
computing mechanism. 

Aided Tracking 

Before leaving this particular discussion of the 
basic sight equation, it may be appropriate to show 
how it is connected to tracking mechanisms. 2 There 
are three principal methods of relating handlebar 
motion to the motion of a gun. These are (1) direct 
tracking in which the angle 7 through which the gun 
is displaced is directly proportional to the angular dis¬ 
placement fx of the handlebars (7 = cm) , (2) velocity 
tracking in which the velocity with which the guns will 
move is proportional to the angular displacement of 
the handlebars (7 = c^tx), and (3) aided tracking in 
which a displacement of the handlebars simultane¬ 
ously displaces the guns and gives them a velocity 
(7 = cm + C2/x). Type 2 is realized by a motor 
whose speed depends on displacement of the control. 
Type 3 implies an exponential smoothing mechanism, 
since with or 1 and c\ the respective analogues of a 
fixed t m and a sight constant a, the theory of the 
aided tracking equation follows that of equation (1). 

Investigations of aided tracking 136 have generally 
shown that aided tracking gives better results than 
do the other types. However, no optimum value for 
ci/c 2 , i.e., the analogue of at m , has been arrived at. 
There is little to distinguish results for values within 
the range 0.2 to 0.8. 

5.3.5 The Calibration Concept 

It has been emphasized that t m , the time-of-flight 
multiplier, is to be chosen to optimize the perform¬ 
ance of the sight over some class of tactical circum¬ 
stances. To illustrate this philosophy, consider the 
set of target paths which sensibly resemble pure 
pursuit curves. The deflection to take against a 
pure pursuit is by equation (1) of Chapter 4 

. . 7 v ° . 

A = Kq sin r 

Vo 

7.1 1 v T 

2 v 0 

Consequently, if t m were given the value 

_ 
tm — 

<7 

correct, shooting against pursuit curves would occur. 


However this simple procedure would not eliminate 
the time lag in lead computation nor would it allow 
for the fact that the actual sight will subtract from 
the kinematic lead that it produces, a lateral ballistic 
allowance /3. To take these two facts into account 
start with the complete basic equation 

at m \ + X = t m <7 — /3, (8) 

where the lateral ballistic deflection /3 is given ap¬ 
proximately, following equation (11) of Chapter 1, by 

= crpvo sin 7, 

where c is a constant and r is present range. Now 
solve equation (8) for t m} using the correct total lead 
A for a pure pursuit curve instead of A. Then 


A + ff 
a — a A 


(9) 


This ‘ ‘forcing” of the differential equation means 
that the sight’s total steady state solution must give 
the correct deflection against a pure pursuit curve. 
The procedure 41 is equivalent to selecting t m such 
that the sight’s lagged kinematic deflection when 
combined with the sight’s ballistic deflection, what¬ 
ever it may be, will yield the correct lead A. 

To reduce equation (9) to a form suitable for cal¬ 
culation, the expressions 


r<r = Vg sin <r, 

k 0 VG cos a ■ a 

A = - 1 

Vo 

sin 7 = sin a + A cos a 


may be used. Then t m is given by 

t m k 0 + crp(v0 + k 0 v G cos a) 

~ = -T- {<? = r)- (10) 

r Vo — ak 0 v G cos <7 


The actual sight mechanisms have been kept 
simple by using a circuit that computes t m as a 
function of present range r, and altitude (relative air 
density p) only. Hence the variation of equation 
(10) with target and mount speeds, and with angle 
off cr are not taken into account. In particular, 
equation (10) is definitely asymmetric with respect 
to the beam (0- = 90°). There are three sources of 
the asymmetry (1) k 0 , when given its exact value 
from equation (1) of Chapter 4, is seen to be larger in 
the forward hemisphere because t f and q are both 
smaller (because of the shorter future range), (2) sin 7 
> sin a in the forward hemisphere and so the trail 
allowance is greater for a given angle a in the forward 
hemisphere than it is for that angle at the rear (in 
the rear hemisphere sin 7 < sin a), and (3) since A 


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BASIC THEORY OF LEAD COMPUTING SIGHTS 


65 


increases in the forward hemisphere (A decreases in 
the rear hemisphere), the time lag does not act to 
counterbalance the first two factors. In sum, the 
total lead put out by the sight is 
d , . 

X = t m a - at m — (t m <r) ~ /?• 
at 

If the t m used is that appropriate to the beam 
(* = 90°), 

tm k 0 
~ = ~ + cpr, 
r Vq 

then in the forward hemisphere t m cr is too small, and 
both the delay term and the ballistic term subtract 
still more. The sight underleads. In the rear hemi¬ 
sphere t m dr is too big, the delay term adds somewhat 
more and the ballistic term does not subtract as 
much as it should. The sight overleads. 

In spite of the arguments for making t m vary with 
cr, vt, and Vg — as well as with r and p — enthusiasm 
for sight refinement must keep in mind that such 
elaboration may again be pointless in view of the 
errors in supplying inputs, and may give excellent 
results for the special class of curves assumed in the 
calibration but deteriorate on other types of target 
paths. A suggested calibration 41 (using averages 
over v G , v t , and a) is, for a = 0.43, API M8 ammuni¬ 
tion (tf 0 = 2870, c 5 = 0.440), and ballistic deflection 
constant b c = 0.0025; 

P r 0 3 6 9 (hundreds of yards) 

t m /r 0.0968 0.1036 0.1101 0.1164 (seconds per hundreds 

of yards) 

Figure 4 gives more details of the calibration. 

5.3.6 Class B Errors in General 

The Class B errors of a fire control system were 
defined in Section 4.5.4 as those arising when a sight 
constructed exactly in accordance with design is 
operated perfectly with no dispersion. To assess 
these errors analytically for a given type of course, 
the calibration in the sense of Section 5.3.5 must be 
known. Early estimates 3 of Class B errors were 
made under the assumption that t m w r as the actual 
time of flight over present range as obtained from 
ballistic tables. From the current point of view it 
is more significant to proceed in two directions. 


b c = 0.003 is the actual value if the future range were r 
(caliber 0.50 API M8 ammunition). The use of c = 0.0025 
takes into account the fact that future range should be used 
in assessing ballistic deflection, and this range is about 85 per 
cent of r for pursuit curves. 4 


(1) Under perfect calibration in the sense of equation 
(10), hov r does the sight perform against target paths 
other than those assumed in the calibration, e.g., 
rectilinear paths? (2) Undef an actual averaged cali¬ 
bration, how does the sight perform against the 
courses of the calibration? 

Against a straight-line course 

. Vg . Vt 

sin A = — sin a — q — sin a 

v Q Vo 

by equation (1) of Chapter 2, and 

ra = Vg sin a — Vt sin a 

by equation (9) of Chapter 2. One could get the 
optimum calibration against straight-line courses by 
inserting these expressions in equation (9). Alter- 



0 2 4 6 8 10 

pr (HUNDREDS OF YARDS) 

Figure 4. Calibration of a lead computing sight when 
used against pursuit curves. (Upper family: a = 0°; 
lower family: o-=180°.) 

natively, as representative of direct calculation, con¬ 
sider a special case of support fire against a rectilinear 
company front attack (Section 4.6.1). Using one 
hundred yards and seconds as units, choose vg = 0.8, 
VT = 2.2, o- = 150°, a = 30°, r = 6, p = 0.5, API 
M8 ammunition. A good value for q is 1.066, so 
that the correct total lead A = 81 milliradians. For 
these conditions a sight calibrated perfectly against 
pure pursuit (k 0 = 0.89) would use t m = 0.586 sec. 
Since the tracking rate is cf = 116.7 milliradians per 
second, the kinematic lead produced is about 68 


CONFIDENTIAL 














66 


LEAD COMPUTING SIGHTS 


milliradians. In this case the ballistic deflection will 
be added by the sight. It is /S = 3 milliradians. 
However, since the lead is increasing, the effect of 
lag is to decrease the deflection. This contribution 
[equation (6)] is about (0.43) (0.586) (33.2) = 8 milli¬ 
radians. The total sight error is, therefore, 18 
milliradians. Although this is a serious bias it must 
be kept in mind that dispersion, input errors, and 
Class A errors may still lead to hits. At least the de¬ 
flection is on the right side of the target. (An own- 
speed sight, which can also be regarded as calibrated 
for pure pursuit curves, would give, if used with 
pipper on target, a deflection of 37 milliradians to the 

R 



rear of the target and so would commit a Class B 
error of 118 milliradians.) 

It is appropriate to continue the discussion of 
Class B errors in those sections concerning particular 
sights. 

5.4 MECHANICAL LEAD COMPUTING 
SIGHTS 

5.4.1 Types 

The realizations of the theory of Section 5.3 that 
will be considered here are called mechanical rate 
sights. The computing mechanism consists entirely 
of mechanical elements as opposed to electrical or 
gyroscopic units. These sights form the Sperry 
series: K-3, K-4, K-9, K-12, K-16. The pairings are 
natural since the K-4 and K-16 are lower ball turret 


versions of the K-3 and K-12 respectively. The K-9 
is a layout redesign of the K-3 to make a Martin 
turret installation possible. All the other models go 
in Sperry turrets. Although the K-12 is also an im¬ 
proved version of the pilot model K-3, most of the 
emphasis of this section will be placed on the K-3, 
since this sight has received more study. All models 
are two component sights with separate mechanisms 
to compute the total lead laterally and vertically. 

5 . 4.2 The K-3 Sight 

A general knowledge of the mechanism is useful. 
The Ball-Cage Integrator 

The essential element is the hall-cage integrator 
illustrated in Figure 5. The speed of a point on the 
disk at the cage is zjl. This speed is transmitted to 
the roller which must have a speed h4>. Hence 



and the output angular velocity <t> is proportional to 
the input displacement z . 

Mechanical Exponential Smoothing 

The ball-cage integrator can readily be made into 
an exponential smoothing circuit. 13 In the schematic 


R 



Figure 6. Mechanical exponential smoothing circuit. 


of Figure 6 a differential (see “Ball-Cage Integrator” 
above) has been added. The rotational inputs are 
the elevation E G and roller angle <f>. Through a 
proper choice of gear ratios, the differential adds the 
two inputs algebraically in a desired proportion. 


CONFIDENTIAL 













MECHANICAL LEAD COMPUTING SIGHTS 


67 


Thus 

Then 


Eg = 4> + cz . 


Eg = 4> + cz , 


fr M . 

Eg = T 2 + cz . 
b 

If one takes c c = 1 + a and varies the disk speed so 
that /x/6 = 1/tm, then, calling the rack output z , the 
vertical kinematic lead Xv, one has 

(1 + a) Xv + ~ Xv = Eg . 

This is the required equation, in the vertical com¬ 
ponent, connecting gun rate to deflection. 

Another ball-cage integrator is used to make the 
disk speed vary as the reciprocal of the time-of-flight 


DISK OF ELEVATION 
SMOOTHING CIRCUIT 



Figure 7. Method of making disk speed depend 
on 1 /t m . 


multiplier. This is shown in Figure 7. In this appli¬ 
cation, the ball-cage unit is used only as a device to 
transform a displacement into an angular velocity. 
It is evident that as the range approaches zero, l/t m 
approaches infinity. Hence the mechanism must 
abandon its computation at a point for which the 
speed would be excessive. d 

Lateral Circuit 

Near the zenith, a small change in target position 
leads to a large change in the azimuth of the target. 
Hence lateral deflection rather than azimuth deflec- 

c The gear ratios are chosen so that a = 0.22 for the K-3 
sight. 

d In the K-3, the limiting range for accurate computation 
is 200 yd. Below that range, a speed corresponding to a 
constant value t m = 0.2 sec is operative. 


tion is computed. This is the deflection in a plane 
whose normal is perpendicular to the gun and which 
lies in a vertical plane containing the gun. The 
appropriate gun rate is, therefore, Ag cos Eg, since 
this is the magnitude of the rotational vector directed 
along the normal just described. The lateral circuit 
will be complicated by the required multiplication by 
cos E g . According to Figure 8, this multiplication 


Oe 


G 

CAM 


H 


D DIFFERENTIAL 

3= “ 




C.i A( 


RACK AND 
PINION 


K 




Figure 8. Lateral smoothing circuit. 


is achieved by yet another ball-cage integrator and 
a cam. The output of the first roller is 

D = Ag cos E g , 
and at the differential 

D = (1 + cl)Xl + 

D = (1 +o)Xl + ^X l , 

Cm 

so that 

(1 + cl)\l Xl = Ag cos Eg . 
tm 


Complete Circuit 

Lateral and vertical ballistics, /3 l and /3 f, are to be 
combined with Xl and X v just before the lateral and 
vertical mirrors of the optical system are rotated. 
Two-dimensional cams are used to put in ballistics. 
(Such cams are irregular cylinders. By rotating such 
a cylinder, one input is made, and by displacing a 
follower along it a second input is possible.) Ballistic 
deflections actually depend on range, azimuth, ele¬ 
vation, altitude, and mount velocity. With two- 
dimensional cams average values for certain variables 
must be selected.® The lateral deflection is taken 


e For the K-3, the average values used for /8 L are altitude 
= 18,000 ft and IAS = 200 mph; for ft v , an average range 
of 800 yd is used. 


CONFIDENTIAL 






























68 


LEAD COMPUTING SIGHTS 


to depend only on range and A G) and Pv to depend 
on E g and A G . The complete sight schematic is 
formed in Figure 9. 


FRAMED TAR6ET LATERAL 



Figure 9. K-3 sight schematic. (Courtesy of Sperry 
Gyroscope Company.) 


5.4.3 The K-12 Sight 

Mechanically, the K-12 sight differs rather radi¬ 
cally from the K-3. Only two ball-cage integrators 
are used so that the kinematic deflection outputs are 
X L/tm and \v/t m . Similarly the outputs of the ballistic 
cams are pL/t m and Pv/t m . A linkage adder combines 
kinematic and ballistic deflections, and then t m is 
multiplied out by a linkage multiplier (which is a 
level with a variable fulcrum positioned according to 
l m ). Linkages also permit the ballistic deflection to 
depend on all five variables rather than two. Caliber 
0.50 API M8 ballistics are used, a is changed from 
0.22 to 0.37, and it is possible to switch from a t m 
appropriate to straight-line courses to 0.90£ m which 
fits pursuit curves. In the mirror system the lateral 
deflection is taken first, and the mirrors are per¬ 
pendicular rather than parallel as in the K-3, im¬ 
proving the system performance by minimizing the 
error due to gun roll. (See Section 2.4.2.) Range is 


controlled by a button operated motor, giving 
velocity ranging instead of the K-3 direct ranging 
(see “Aided Tracking” in Section 5.3.4). The reticle 
consists of 12 dots (instead of a gate as in the K-3). 
Input intervals are: range 200 to 1,500 yd; target 
dimension 30 to 60 ft; IAS 100 to 450 mph; altitude 
0 to 40,000 ft; gun rates up to 250 milliradians per 
sec; elevation up to 85°. 71 

5.4.4 Complete Sight Equations 
and Origin of Class B Errors 

The Class B errors (Section 4.5.4) of an actual 
sight are attributed not only to approximations in 
the original theory but also to compromises made in 
the design and to the peculiarities of mechanization. 
Consequently, with a real sight, in discussing errors 
of this type it is customary to say that the sight 
behaves exactly according to blueprint specifications. 
Everything is to be included except the statistical 
features of dispersion, manufacturing and function¬ 
ing variations, and input errors. 

For the K-3 sight the kinematic deflections are 
computed by 

(1 + a)t m \L + Xl = t m A G cos E g 
(1 -f- a)t m \v + Xf == tmE G , 

and these are combined linearly with the ballistic 
deflections p l and Pv to give the total deflections 

Al = X L - Pl, Pl = Cr sin A G + D 

A F = Xf — Pv, Pv = F sin E G cos A G (12) 

— G cos E g — H , 

where C, F, and G are constants. In working with 
these equations the blueprint t m , Pl, and Pv are to be 
used. To complete the description of the sight the 
approximate mirror equations 74 are 

A g = A + Al sec E + M L 
E g = E + Af + My, 

where 

Ml= -secE-AiAv ,.o\ 

My = $Ai(l - 2 Af) tan E. 

These describe how the optical system relates the 
azimuth and elevation of the sight line, A and E, to 
the angular coordinates of the gun, A G and E G . 

It is readily seen how errors other than those of the 
underlying theory (Section 5.3.6) must arise. The 
complete equations for kinematic deflection 206 should 
contain an additional term on the right of equation 
(11) to account for gun roll. (See Section 2.4.2.) 


CONFIDENTIAL 





































GYROSCOPIC LEAD COMPUTING SIGHTS 


69 


Next, /3 l and Pv from equation (12) do not use p, vg, 
E g , and p, Vg, r, respectively, as they should. Finally, 
in regard to equation (13), the error terms M L and 
M v show how the mirror system fails to do its job. 
And there are subtler phenomena than these. Near 
the zenith, lead is changing rapidly from one com¬ 
ponent to another, and the ballistics are also inter¬ 
changing. The lag effect of the exponential circuit is 
enhanced, and spurious terms can be introduced into 
the lead computations. Again, there are feedbacks 
from the ballistics and from the mirror system. To 
understand this phenomenon, simplify the above 
mathematical description of the sight to read 

(1 + a)t m \ + X = t m y (smoothing) 

A = A — p (ballistics) 

7 = cr + A + M (mirror). 

These equations may be combined to yield 

atmk -f- A = t m a — p — (1 -f- a)t m P + t m M. 

The last two terms are the ballistic and mirror feed¬ 
backs and are, of course, extraneous. 

The analytical assessment of the Class B errors of 
a sight determines the total errors that the sight 
makes against some class of courses, and decomposes 
the total error into its several causes. This program 
has been carried out for the K-3. 21 One may take a 
class of, say, pure pursuit curves for which the exact 
deflection is known as a function of position. The 
tracking rates and ranges are also known as functions 
of position. The system (11), (12), and (13) is then 
to be solved for the steady state A l and A v- Since 
the solution is in symbolic terms, the identity of each 
part of the total error can be established. The proc¬ 
ess is lengthy and involves a sequence of approxima¬ 
tions with the result that the final values have a 
probable accuracy of 3 milliradians. 

In summary of this computation, the lateral error 
is given by 

(A G — A* g ) cos E = a sin A + b sin 2 A 

+ c sin 2A cos A, (14) 

where a, b, c are not constants. However a is inde¬ 
pendent of elevation, and b and c are independent of 
target speed. The vertical error is given by 

E g — E* G = d cos A + e sin 2 A 

+ / sin 2 A cos A + g, (15) 

where e, f, and g are independent of target speed, and 
e and / are independent of range. 

The lateral error may be interpreted as follows. 
(1) a sin A is attributed to the curvature of the tar¬ 


get’s path — it causes the sight to lead 10 per cent in 
excess when t m is time of flight over present range; 
(2) b sin 2A is due to gun ^oll, or, if one does not 
wish to elaborate the basic equations (11) to include 
this, is due to the unsuitability of the mirror system; 
and (3) c sin 2 A cos A represents the combined effect 
of delay, feedback, and interchange. Similarly, the 
vertical error may be interpreted as follows. (1) d 
cos A is again in this component, the error arising 
from the neglect of the curvature effect in choosing 
t m , although at short ranges the incorrect gravity 
drop (by equation (12) independent of range) is so 
large that it compensates for this error; (2) e sin 2 A is 
composed of errors due to gun roll, delay, feedback, 
and interchange/ (3) the third term is principally 
caused by the mirror system and its feedbacks; and 
(4) g represents a failure to make gravity drop vary 
with range. 

Detailed tabulations of Class B errors for this sight 
have been made. 14 ’ 21 The cellular array of Figure 10 
is illustrative of the results. 


V B = 225 M PH H* 20,000 FT PLANE OF ACTION 



OUTER NUMBER IN EACH CELL IS ERROR PARALLEL ^TO PLANE OF 
INNE.R .. PERPENDICULARjACTION IN 

Jmilliradians 

Figure 10. Class B errors of K-3 against pure pursuit 
curves (milliradians). 

5.5 GYROSCOPIC LEAD COMPUTING 
SIGHTS 

5.5.1 General Nature of a Single 
Gyro Sight 

Mechanical lead computing sights are essentially 
computing machines or brains which work through 
certain formulas. Gyroscopic sights, which utilize 

f Typically at A = 90° and E = 45°, the error of 28 
milliradians for this term breaks down to 15, 6, 4, 3 milli¬ 
radians respectively. 


CONFIDENTIAL 







70 


LEAD COMPUTING SIGHTS 


the properties of a gyroscope, which employ eddy 
currents, and which construct physically (rather than 
calculate mathematically) the required deflection, are 
more like sentient beings. A much more elementary 
example of this point occurred in the comparison of 
vector and algebraic own-speed sights (Section 4.5.3). 

A significant advantage of a gyro sight is that it 
will measure the true rate of rotation in the air mass 
of the line connecting gun to target. In certain in¬ 
stances, the tracking may actually be done by the 
motion of the firing aircraft. For example, in firing 
at a fixed ground object from an aircraft making a 
pylon turn on that target, the gunner has nothing to 
do. The gyro sight will establish the angular rate 
a, = VG /r, will combine this with a time of flight 
t m = qr/v o to give a kinematic deflection, behind the 
target, of qvg/v o and will subtract from this the lateral 
ballistic allowance which will decrease qvg/v o to full 
own-speed allowance vq/v o which is the required de¬ 
flection. A mechanical sight, under these circum¬ 
stances, could supply only a trail allowance ahead 
of the target. In bombers, this independence from 
aircraft motion in obtaining correct deflection is im¬ 
portant, but in fighters it is imperative. 

The sight considered in this section uses a single 
flexibly mounted gyro. (Certain sights used for anti¬ 
aircraft purposes and for fighters — the Draper- 
Davis sight and the German EZ42 — employ two 
suitably constrained gyros and solve the problem 
componentially.) The single gyro sight gives the total 
kinematic deflection along the relative path of the 
target regardless of the direction of that path. 

The gyro sight consists essentially of (1) an elec- 
tromagnetically controlled gyroscope which measures 
kinematic deflection and also introduces ballistic cor¬ 
rections, and (2) an optical system which establishes 
the line of sight and introduces the sight parameter. 
We shall consider constructional details 191 but 
slightly. In operation, the gunner presets altitude, 
airspeed, and target span. During an attack, he 
tracks with pipper on target, keeps the target framed 
by the reticle pattern, and fires. Azimuth and eleva¬ 
tion of the guns are automatically picked off the 
turret gear trains for use in determining ballistic 
corrections. 

The single gyro sight for a turret is called (1) the 
Mark II C by the British, who originated the design, 
(2) the Mark 18 by the U. S. Navy, which modified 
the design for U. S. ammunition, and (3) the K-15 by 
U. S. Army Air Forces. The fighter version (Mark 
21, Mark 23, K-14) is discussed in Section 5.6. 


5.5.2 Method of Producing Kinematic 
Deflection 

To arrive at the basic principles of the sight, it may 
first be explained how kinematic deflection is pro¬ 
duced. 72, 73 Suppose that a gyroscope is mounted on a 
gun so that the spin axis is parallel to the bore axis 
of that gun, and so that the universal joint mounting 
(Figure 11) at 0 is the center of mass of the gyro 



system and is also the point about which the gun 
rotates. Since the mounting is universal, as the gun 
is moved about 0 the gyro axis will remain pointing 
in its original direction in space, this being a basic 
property of gyroscopes (used for example in the gyro 
compass and flight-attitude gyro). But as shown in 
Figure 11, in tracking a target we require the gyro 
axis, if we consider that axis as the line of sight, to 
lag behind the guns an appropriate amount to gener¬ 
ate deflection. This can be done only by making the 
gyro precess, since, in tracking, the line of sight must 
stay on target. To make the gyro precess in the 
plane of the tracking, a force F must be applied at a 
point such as A and directed at right angles to the 
plane of the tracking. (Witness the dip of a single 
engine aircraft in making a horizontal turn due to 
the action of the propeller and the horizontal forces 
on the tail surfaces.) 

If I is the moment of inertia of the gyro, if T is the 
applied torque (equal to the force F times the dis¬ 
tance l = OA), and if ft is the angular velocity of 
spin, then, by a familiar formula of mechanics, the 
precessional rate a is given by 
1 T 

G ~ I ft ‘ 

Since moment of inertia, spin, and the distance OA 
are normally fixed, a change in the rate of precession 
requires a change in the applied force F. Hence 
write 

<j = CiF, 


CONFIDENTIAL 











GYROSCOPIC LEAD COMPUTING SIGHTS 


71 


where Ci is the constant l/Itt. If a and y are meas¬ 
ured from some reference line fixed in space, 8 e.g., 
pointed at a star, then, if the force F could be made 
proportional to the angular separation of gun and 
sight lines, we would have 

o■ = CiC2(y — a). 

The force F could be made proportional to the angle 
7 — a by attaching a spring from A to B and relying 
on Hooke’s law. F would then be right in amount 
but wrong in direction. If c 2 could be made inversely 
proportional to a time-of-flight multiplier t m (vari¬ 
able spring stiffness), we would have 


CiCz 

o- = — (7 


<7). 


Finally since 7 — a is the lead X, a design such that 
C1C3 = 1 would lead to 

X t m (T , 

and kinematic deflection would be produced. The 
only things wrong with the scheme are (1) a spring 
mechanism would cause precession at right angles 
to the plane of tracking, and (2) no smoothing is 
produced, i.e., a = 0. The first point can be met 
by an ingenious application of electrical eddy cur¬ 
rents. This is described in Section 5.5.3. Recalling 
Section 5.3.3, smoothing can be achieved not by 
using the spin axis as the line of sight, but by keep¬ 
ing the sight axis a fixed proportionate distance be¬ 
tween spin axis and gun axis. It is one function of 
the optical system to do this. The method is given 
in Section 5.5.4. 


5.5.3 The Physics of the Gyro 
System 

Instead of a cylindrical disk, the actual gyro of 
the sight consists of a copper dome, a flat circular 
mirror, and a short connecting axle. This system 
revolves at approximately 3,000 revolutions per min¬ 
ute about a universal mounting to be considered 
at greater length below. The cap of this toadstool 
moves between the two poles of an electromagnet 
whose current can be varied through a variable re¬ 
sistance. This arrangement is schematized in Figure 
12 . 

The strength H of the magnetic field between the 
poles is 

H = cd 


g See Figure 1. 


s — '^mzzzi' 

GUN 


ELECTROMAGNET 



SIGHT HEAD 

Figure 12 . Gyro schematic. 

where c 4 is a constant, and i is the current in the 
coils. The lines of magnetic force pass through the 
dome, intersecting it in a circular area centered at 
A. The area of one instant is immediately replaced 
by another since the dome is spinning. Eddy cur¬ 
rents are induced in the dome, since a circular 
annulus of the dome can be thought of as a coil of 
wire moving across a magnetic field. The current j 
induced in such a coil is given by 

j = c b Hv, 

where C5 is a constant and v is the velocity with 
which the coil moves. 

The effect of these eddy currents can be made out 
heuristically by an examination of Figure 13. The 
induced eddy currents set up the equivalent of two 
magnets in the instantaneous dome segment, with S 
and N poles on the right and N and S poles on the 
left. The resulting four forces all act to oppose the 
motion with a force 

F = CeHj, 

where c 6 is a constant. The force F tries to slow the 
dome down. But the constant-speed motor supplies 
a couple with a force F/2 at A (Figure 12) opposing 
F and a force F/2 at A' in the same direction as F . 
Hence the speed of rotation is maintained and an 


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72 


LEAD COMPUTING SIGHTS 


unbalanced force (F — F/2) + F/2 = F is left 
which is directed down into the paper in Figure 12. 
Hence precession will occur in the plane of the spin 
and bore axes as it should. 



Under tracking, then, as the bore axis is displaced, 
the spin axis tries to remain fixed, but the displace¬ 
ment causes a force F which leads to precession of 
the spin axis toward the bore axis. If the gun keeps 
rotating, the spin axis does not catch up and the two 
axes revolve with an angular displacement between 
them. 

The applied torque, assuming that the moment 

arm OY = OA = l, is 

^ 7 2 (22) 2 

T - Zc 6 c 5 c 4 v. 

And if r is the (variable) distance YA we have 
v = ril = l\£l, 

where X = y — a. But the precessional rate 
.IT c 
' r ‘/Q = R°- X ’ 

where (16) 

C6C 5 (22Zc 4 ) 2 
c - j 

Finally, 11 if R 2 /c is chosen to be t m , we have 
X = t m <j . 


h In connection with c it should be observed that, since the 
22-volt power supply is from the aircraft, a p per cent error 
in regulation means a 2 p per cent error in the deflection out¬ 
put. 


5.5.4 Optical System and Optical Dip 

The optical system must introduce smoothing. 
The system is schematized in Figure 14. The reticle 
gate consists simply of one flat disk provided with a 
central hole and six radial slits placed against an¬ 
other disk provided with six curved slits. The reticle 
pattern consists, as a result, of a pipper and six 
diamond dots (due to the overlap of slits). By ro¬ 
tating a disk with foot pedals, the size of the pattern 
is changed, a target may be framed and present 
range supplied to the range coils. 



The optical distance from reticle gate to lens is 
the focal length / (7.078 in.) since this is a colli¬ 
mating system. The optical distance from gyro 
mirror to lens is the image distance u (4.578 in.). 
The distance v to the (virtual) object is given by 
1 1 _ _ 1 _ 

u v f 

Consider a ray emanating from the image point 
(gyro mirror) and making a small angle a with the 
axis of the lens. Upon refraction by the lens the 
exit ray makes a (smaller) angle 1 /3 with the axis 
of the lens determined by 

' tan 0 u 
tan a v 


Combination of these relations yields 
f — u 

tan j8 = —~— tan a = 0.353 tan a . 

This result is only approximate. If thick lens theory 
is used, 75 it is found, more exactly, that 


0 = 0.353a - 0.0895a 3 + 1.185a 5 . 


5 Do not confuse this angle /3 with ballistic deflection, also 
frequently called 0. 


CONFIDENTIAL 


























GYROSCOPIC LEAD COMPUTING SIGHTS 


73 


This point is labored since it will be seen immediately 
that the sight parameter a will not be constant but 
will vary in a peculiar way depending on the amount 
of deflection. 

Suppose now that the gyro mirror is rotated in ele¬ 
vation through an angle \y. (The design is such that 
0 is effectively in the mirror. See Figure 16.) The 
reflected ray from the gyro mirror is then displaced 
from its original position by twice this angle. Upon 
hitting the fixed mirror the reflected angle remains 
2\f so that the entrant ray to the lens makes this 
angle with the lens axis, i.e., a = 2\ v . Hence the 
exit ray from the lens becomes a reflected ray from 
the combining glass and makes an angle 

0 = 0.706X f - 0.716\f + 37.92\£ 

with the original position. Consequently the ratio 
of the angle from gyro axis to sight axis to the angle 
from sight axis to gun axis is 

Xf — 0 
a =-• 

P 

It is more important to consider a rotation of the 
gyro mirror in azimuth, since in this case not only is 
a a function of the deflection Xa but also a spurious 
elevation angle is introduced leading to the phenom¬ 
enon of optical dip, which is an error of the sight. 
The doubling principle used above depended on a 
rotation of the mirror about an axis perpendicular 
to the plane of incidence. For rotation around some 
other axis doubling is not quite accurate. 18 This may 
be explained crudely, for an azimuth rotation, as 
follows. In Figure 15, the deflected ray to the lens 
has farther to go to get to the lens, and climbing at 
17° it hits the lens a small distance above the central 
line of the lens. To the combining glass of the sight 
a dip has apparently been introduced. The glass 
thinks that the gyro has also changed in elevation. 
Furthermore, instead of full doubling, a factor 2 cos 
17° appears. In fact, a simple trigonometric analysis 
of Figure 15 shows that for a displacement of the 
gyro axis, in azimuth only, of \a radians, the sight 
axis will move in azimuth by 

(0.353)2 cos 17° \a radians 
and will always depress in elevation by 
(0.353)2 sin 17° X| radians. 

First approximation thin lens theory is used, i.e., 
0 = 0.353a.) In general, dip may be made out with 
sufficient accuracy by this approximate theory. 

If the gyro axis is displaced from the gun axis by 


angles \a and Xf in azimuth and elevation, then the 
sight line is displaced from the gun axis by 

0.675Xa + 0.206XaXf radians, 

0.706Xf — 0.1975Xa radians 

in azimuth and elevation, respectively. It is evident 
that dip is due to the design necessity of having a ray 
of light from the reticle strike the gyro mirror at 17°. 1 



To get at the way in which a changes, the exact 
theory may be used. 75 * 135 The variation in a is given 
by the formula 

—= 0.706<4, - 0.716d’X i 2 * + 37.92d£x 4 , 

1 — a 

where X is the angle between gyro and gun axis and 
d m is the distortion factor 

d m = Vl — sin 2 17° cos 2 \p, 


i In certain German single gyro sights, this is not true. The 

fixed mirror is half-silvered and dropped down to the OY 

axis (see Figure 14), and the reticle system is raised to this 

axis so that it is behind the fixed mirror. See Section 5.7.3. 


CONFIDENTIAL 














74 


LEAD COMPUTING SIGHTS 


i p being the angle between the horizontal plane and 
the plane of the gun and gyro axes. The value us¬ 
ually assumed for a is 0.43, but it can readily vary 
from 0.41 to 0.48. This variation has a negligible 
effect on sight performance. 

5.5.5 Mechanical Details 

It is to the point to consider actual mechanical de¬ 
tails since such knowledge leads to an understanding 
of the limitations inherent in mechanisms of this 
order and is of assistance in the general design of 
new equipment. 

Hooke’s Joint 

The actual gyro system employs a Hooke’s joint k 
for its universal mounting. As indicated in Figure 
16, the gimbal is a flat cross of metal whose input arm 



AB = INPUT ARM 
C D = OUTPUT ARM 

Figure 16. Actual gyro unit. 


is pivoted to the driving pulley, while the rotor 
assembly is hung on the output arm. The pulley is 
placed in ball bearings in a panel connected to the 
sight head and in a vertical plane perpendicular to the 
bore axis of the gun. The axle can assume any angle 
with respect to the pulley by rocking about the four 
ends of the gimbal cross, and rotation is maintained 
at any such angle. This implies that the output arm 
must bob up and down 3,000 times per minute. The 

k The proper oiling of this joint is of the utmost importance. 


result of this gimbal inertia is a slight torque tending 
to push the axle still farther from the vertical. This 
tends to cause a conical precession of the axle. (The 
cross-wind force on a bullet which acts radially out¬ 
ward in the instantaneous plane of yaw is analogous.) 
Another torque which has a similar effect is caused by 
the rapid spinning of the mirror while in a cocked 
position. The asymmetry in the sight head leads to 
low air pressure on one side of the mirror and high 
pressure at the other. This is the Venturi torque. 
Additional torques are due to air drag and frictional 
resistances , and, of course, the driving torque as modi¬ 
fied by the joint. These five torques are extraneous 
in the sense that the only torque we really want is the 
one due to eddy currents in the dome. Evidently, 
the anafysis of the gyro system is complicated. The 
results 16 will not even be quoted. From the point 
of view of ultimate consideration of Class B errors, 
however, the conical precession referred to above im¬ 
plies that a displaced gyro axis does not return to the 
gun axis along a straight line. This leads to a dip 
additional to the optical dip of Section 5.5.4. Since 
its magnitude is approximately 

0.025t*<7 milliradian 

it follows that it can be positive or negative depend¬ 
ing on the direction of tracking and so may partially 
balance, or augment, the optical dip. 

Electromagnetic System 
The electromagnetic system actually uses four sets 
of poles instead of the single pair of Figure 12. The 
range coils are not wound individually around each 
pole, but are wound around the entire unit of four. 
The fields between the several pairs are the same in 
magnitude and sense. Ballistic deflections are taken 
into account in a simple fashion. In addition to the 
enveloping range coil, each pair of opposite poles is 
wound with a separate ballistic coil (Figure 17). 
The fields produced are opposite by pairs because of 
the winding. Referring to Figure 3 of Chapter 1, it 
is seen that the two sets of ballistic coils are needed 
to produce the appropriate lateral and vertical com¬ 
ponents of the trail IF. These components are, re¬ 
spectively, 

W sin A G and W cos A G sin E G . 

An additional coil, on P x and P 3 only, is needed for 
the gravity allowance. Under these four windings 
the four electromagnets become equivalent to one 
placed at some point between them, a magnetic 
center of gravity. The single point at which the total 


CONFIDENTIAL 










GYROSCOPIC LEAD COMPUTING SIGHTS 


75 


magnetic force seems to act is called the magnetic 
center , and the line from this point to the Hooke’s 
joint is the magnetic axis. When ballistic currents 
flow, the ballistic fields, being opposite in sense, cause 
the magnetic center to shift. To the gyro the mag¬ 
netic axis now plays the role of the gun axis and it is 
with respect to the magnetic axis that kinematie de¬ 
flection is taken by the gyro. In this physical fashion 
the gyroscope constructs the difference between 
kinematic and ballistic deflections and properly posi¬ 
tions the gun. 



Figure 17. Range and ballistic coils. 

Class B errors can occur because of the electro¬ 
magnetic system. The lateral ballistic coils not only 
pull the magnetic center over as they should but they 
also deflect it slightly in elevation, introducing still a 
third dip. Again, as might be expected, the ballistic 
coils also act in a small way as range coils, changing 
t m slightly. Finally, an important error could be in¬ 
troduced due to the variability of conductivity of the 
dome with temperature which changes radically from 
sea level to high altitude. (Lead will change by 0.4 
per cent per degree C change in temperature.) Since 
the altitude input changes resistances, it follows that 
these resistances must be so chosen that the resulting 
currents will compensate for this effect. 

Electrical Circuit 

The electrical circuit must take the inputs of range, 
position of target, etc., and supply suitable currents 
to the coils of the electromagnets. To translate 


properly the required formulas, nonlinear resistances 
and attenuating (multiplying) circuits are used. 
Suppose, in Figure 18, that the variable resistance Ri 



Riii = Ran 



ii = ^ = kq»qi 
ki = k 

Figure 18. Multiplying circuit. 


is set by a dial number q h and the variable resistances 
R 2 and R 3 are set simultaneously by the dial number 
q 2 . The problem is to design variable nonlinear re¬ 
sistances Rij R 2 , and R 3} and to choose a fixed re¬ 
sistance Ri so that the current in the box is 


ii = Kq x q 2 . 

Now the resistance from A to B will always be con¬ 
stant for all values of q 2 if we choose R 2 , R 3f Ri such 
that 


c = R 2 -\- 


R3R4 

R 3 ~b Ri 


) 


where c is a constant. If R 3 is chosen so that 


Rz 

Rz ~b R\ 

and if Ri is chosen so that 


kq 2 


E K 
Ri + c = k q ' 


we will have, as required, 

ii = Kqiq 2) 


where k, c, and Ri are at our disposal. 

Using this principle, the complete circuit is laid out 
as Figure 19. It is evident that t m is taken to depend 
only on range and altitude, and is made out by a 
simple summing of variable range and altitude 
resistances. So values of R 2 and H 3 must be de¬ 
termined experimentally to give optimum results 
and errors can naturally be expected. 196 The triple 
branching in the trail circuit corresponds to the 


CONFIDENTIAL 




























76 


LEAD COMPUTING SIGHTS 


RANGE CIRCUIT 



BY ONE DIAL 


R,,R 2 - RANGE RESISTANCES 
H, ,H 2 ,H 3 - ALTITUDE RESISTANCES 

A, ,A 2 - AIRSPEED RESISTANCES E^CH PA UR OPERATED 

Z, ,Z 2 ,Z 3 “AZIMUTH RESISTANCES 
E| ,E 2 ,E g ” ELEVATION RESISTANCES^ 

R,Ra,Re,R$ ARE THE COILS OF THE ELECTROMAGNETS 


) 


r E -GROUND TEMPERATURE COMPENSATOR 
X - IN OR OUT RESISTANCE (OUT FOR NOSE) 
R f ,R a -fixed RESISTANCES 

Figure 19. Complete electrical circuit. 


multiplication of density, airspeed, and range re¬ 
quired by the expression for trail (Formula (11) of 
Chapter 1). The final internal branchings introduce 
multiplication by cos A(?-sin E G and sin A G , re¬ 
spectively. Gravity drop depends on range and cos 
E g but not on altitude or airspeed. 

5.5.6 Specific Origin of Class B 
Errors 

In getting at the Class B errors of the gyro sight a 
natural first step is to determine its behavior in sym¬ 
bolic language against a target on a general space 


path and with an arbitrary range r and tracking 
rate <j. 16> 76-78 This requires a complete theoretical 
analysis of the instrument. Given such an analysis 
it is possible (1) to determine what design constants 
should be used to effect an optimum calibration, 
(2) to test the performance of the sight in a given 
class of tactical circumstances, and (3) to assign parts 
of the total error to their particular causes. In 
illustrating this program briefly, we shall restrict 
quotations to those pertinent to motion of target 
and gun mount in a horizontal plane. 

On the surface of a unit sphere let U and V be the 
azimuth and elevation displacements of the gyro axis 


CONFIDENTIAL 













































GYROSCOPIC LEAD COMPUTING SIGHTS 


77 


from the gun axis which is assumed to lie in a hori¬ 
zontal plane. Then, if u and v are the azimuth and 
elevation of the sight axis with respect to the gun 
U — (1 + <h)u V = (1 + (h)v, 

where ai and a 2 vary slightly, depending on w and i; 
(Section 5.5.4). We shall take ai = a 2 = a (con¬ 
stant). For gun rotation in a horizontal plane, the 
general equations become, after linearization, 

au = — (1 -f- a) (p R -j- Dp 2 + At\ + Bt%)u 

— (1 + CI)PgV — cr + CpTi 
(1 + a)v = (1 + a)pou — (1 + cl)(pr + Dp 2 + Bt\ 

+ At\)v + Cpr 2 , 

where p is the range coil current, n and r 2 are bal¬ 
listic coil currents (gravity neglected), A, B,C, D are 
constants determined by the geometric and material 
character of the electromagnetic system, pa is a small 
positive constant (0.02) due to gimbal inertia and air 
currents, and p R is a small positive constant (0.035) 
due to bearing friction and to air drag on the rotor. 
Neglecting only p%, these equations have as steady- 
state solutions (assuming * and range are constant) 

u = 4»i0’ — Pg(1 + a)4uft — ft, 

V = ft + Pg( 1 + d)t m 2{tmia — ft), 

where 


(1 + a){pu + Dp 1 + At\ -+- BtI) ’ 
(1 + cl)(pr + Dp 2 + Bt\ + ^4r2) ’ 

@1 CptmlTi, 

@2 = Cpt m 2T 2 - 


For a horizontal plane, r 2 = 0, since there is no 
vertical component of trail, (ti and r 2 were given 
above because the discussion is also applicable to a 
vertical plane.) Numerically 78 A = 0.26 ,B = 0.18, 
C = 1.374, and D = 20. To get at the character of 
the situation take A = B = 0.23. PuU w = 0.035/p 2 . 
The equations above do not include the effect of 
optical dip. If this effect is superposed, using the 
approximate thin lens results of Section 5.5.4, we 
obtain 

u = O.6750i<r — j5i) 

v = 0.197501* - ft) 2 + 0.0254(ho* - ft), 
where 


0.0354* + 0.7 + 3.5ft 


If there were no bearing friction or air drag the 
term 0.0354* in 4 would be missing. If the trail 
coil currents did not affect time of flight 3.5ft would 
be missing. Then 4 = 4*/0.7, and if the optical 
system were perfect, the number in u, 0.675 (which 
is (0.353)2 cos 17°), would be 0.7 and we would 
have u = t m a — ft i.e., a perfect sight in azimuth. 
Similarly, if the optical system were perfect, the first 
(optical dip term in v would vanish, i.e., sin 0° in¬ 
stead of sin 17° in 0.1975 = (0.353)2 sin 17° cos 17°), 
and if Venturi torque and gimbal inertia were not 
present, the second electromagnetic dip term in v 
would vanish. 

This discussion, however, is academic in the fol¬ 
lowing sense. It assumes that the correct time-of- 
flight multiplier t m is given by t m = 0.035/p 2 . As 
noted earlier, it is necessary to calibrate the simple 
range circuit to give optimum results over a set of 
tactical conditions, such as a family of pursuit 
curves. In doing this it is possible to ameliorate to 
a certain extent the above types of errors, but it will 
not be possible to make t m exactly correct in any 
event. Without giving the numerical values of re¬ 
sistances adopted in the final circuit, the typical size 
of the total lateral and vertical Class B errors can be 
inferred from the summary in Figure 20. 


V 0 * 225 MPH H = 20,000 FT PLANE OF ACTION 

0 HORIZONTAL 



OUTER NUMBER IN EACH CELL IS ERROR PARALLEL 
INNER . PERPENDICULAR 

Figure 20. Class B errors of a gyro sight against 
pure pursuit courses. 


TO PLANE OF 
ACTION IN 
MILLIRADIANS 


It must be expected that an optimum calibration 
for pursuit curves will cause sight performance to de¬ 
teriorate markedly when the sight is used against 
straight-line courses. This is the case. 79 The chart 
of Figure 21 shows by how much the optimum time 
of flight must be increased to give the optimum time 


CONFIDENTIAL 







78 


LEAD COMPUTING SIGHTS 


of flight against straight-line courses. (In each cell 
one number applies to an approach angle of 0° and 
the other to an approach angle of 90° as indicated in 
a typical cell.) These are also approximately the 
percentage errors in lead. 


V c =225 MPH v T = 358 MPH 



PERCENTAGE BY WHICH t m opt (PURSUIT) 

MUST BE INCREASED TO GIVE t m opt (STRAIGHT LINE) 

Figure 21. Relation between pursuit and rectilinear 

calibrations. 

If a blanket percentage increase in t m were decided 
upon, in passing from pursuit courses to rectilinear 
courses, no redesign would be necessary. The intro¬ 
duction of a suitably fictitious target dimension 
would accomplish the purpose. 

5.5.7 Different Type Turrets 

To conclude this section, the effect on the sight de¬ 
sign of a change in the turret type may be considered. 
The classical turret, on which the design discussed 
above is based, positions its guns by rotations in 
azimuth and elevation. 1 As a consequence of this it 
was necessary to decompose ballistic deflection into 
lateral and vertical components. Tracking in an ele¬ 
vated plane of action is not easy, and the tracking 
pattern is frequently stepwise. If, on the other hand, 
a turret is employed in which one rotation is about 
a longitudinal axis of the aircraft (instead of the 
azimuth rotation about a vertical axis) then a plane 


1 Thus, a sight head with one edge perpendicular to the 
bore axis and parallel to the azimuth plane moves so that this 
edge remains parallel to the azimuth plane. For this reason, 
a camera fixed to the guns and recording correct shooting 
against a pursuit curve would show the path of the target, 
on the composite of all frames, as an arch. (Combine this 
motion with Figure 6 of Chapter 4.) 


of action is established in which the guns may make 
a second rotation. (If the elevation rotation is 0 
and the plane of action rotation is /*, then sin 0 = cos 
E g sin 0, sin E G = sin \x cos 0, cos 0 = cos A G cos E G .) 
Tracking should be improved in such a watermelon 
turret. And the gyro sight is now simplified since 
all the trail is in the plane of action. But still a re¬ 
design of the electric circuit is required for good 
results. 80 - 81 

It has also been proposed that a turret be installed 
with its azimuth ring inclined at an angle of 15° with 
the horizontal (to improve aircraft performance). 
Again the sight will supply wrong ballistics unless 
some correction is made. The naive proposal of 
feeding E G — 15° instead of E G actually gives excel¬ 
lent results. 82 

Consideration of the two examples above shows 
the need for the closest integration of armament and 
fire control design. Moreover, since a change in 
muzzle velocity and ballistic coefficient of the ammu¬ 
nition may call for a redesign of a sight, ordnance 
must also work closely with armament and fire 
control if production and utilization schedules are to 
be effective. Otherwise a sequence of modification 
or patching operations becomes necessary. 

5.6 GYROSCOPIC SIGHTS IN FIGHTERS 

5.6.1 Simplifications 

Because of the radical maneuvering of a fighter 
during an attack, it is evident that the only principle 
now at hand on which a sight for a fighter can be 
based is that of the gyroscope. Since the guns are 
pointed by such maneuvering, rates relative to the 
gun mount do not even exist. There is no reason 
why the single gyro sight described in the previous 
section cannot be used immediately by pursuit air¬ 
craft. As a matter of fact it is possible to simplify 
the circuits materially. In the first instance, since 
the classical fighter always fires within a few degrees 
of its direction of motion — the difference being due 
to angle of attack of the guns — it follows that no 
allowance for trail need be made and that the ballistic 
coils and circuits may be deleted. Because of the 
diving and banking it is pointless to leave in the 
gravity coil and circuit since any inputs with refer¬ 
ence to the fighter would be in elevation error. Next, 
it is to be expected that time of flight produced can 
be made quite accurate for two reasons (1) direction 
of fire with respect to the aircraft may be properly 


CONFIDENTIAL 







GYROSCOPIC SIGHTS IN FIGHTERS 


79 


neglected, and (2) calibration for the standard pur¬ 
suit curve attack can be made and will hold, since 
this is the tracking situation with which the fighter 
must contend (the case of fighter versus fighter on a 
curved course is considered in Section 5.6.3). Alto¬ 
gether one expects good results from a gyro sight 
installed in a fighter, particularly in view of large 
size of the target and the high firepower of the fighter. 
The operational problems of tracking and ranging 
are still present, of course. 

5.6.2 Calibration 

To calibrate the fighter sight 41 suppose (1) the 
sight transient has decayed, (2) the target flies a 
straight-line course at constant speed, (3) the bullets 
leave in the direction in which the fighter is flying, 
and (4) ranging and tracking are perfect. API M8 
ballistics will be used (v 0 = 2,870 fps and c 5 = 0.440). 

The correct lead A is then given by equation (3) 
of Chapter 2 and is 

Vt . qvr 

sin A = — sin a = —;-sin a , 

U Vq + V F 

where q , as usual, is Uq/u = ( Vq-\-v f )/u . The time-of- 
flight multiplier t m is obtained by inserting A in the 
equation of the sight. This is the meaning of cali¬ 
bration. We have 

A 

tm "• 7 

a — a A 

By use of evident expressions for a and A this be¬ 
comes 

tm _ _ q _ } 

r ( v 0 — lv F )(l — a ^ Vr CQS a - \ + al(y T cos a + v F ) 

\ Vq ~\~ V F / 

where l = q — 1. Computations based on this 
formula show (1) that t m /r is remarkably insensitive 
to the speed of the fighter, v F , for a = 0.43 (this is 
fortunate since v F is not to be an input), (2) that 
variations in a around 0.43 are quite irrelevant, and 
(3) that increase in target speed magnifies the effect 
of approach angle (but for fast targets one can 
concentrate on tail cone approaches). 

Averaging with respect to those variables which 
cannot be used as inputs, the following table for 
t m opt is to be used. 185 

The time of flight used by the gyro is inversely 
proportional to the square of the current i in the 
range coil (Section 5.5.3). The proportionality factor 
K is, in turn, a function of p, T, a, where T is the 


ambient temperature of the cockpit. It is determined 
experimentally. 186 The design problem, therefore, 
is to choose resistances so that t m oP t is as close to 
K(p, T, <r)/i 2 as possible. The details 185 need not be 
pursued. 


Table 1. Average optimum time of flight for fighter 
gyro sight [t m (r, p) (seconds)]. 


Range r 

Relative air density 

(yards) 

1.0 

0.6 

0.3 

200 

0.220 

0.215 

0.210 

400 

0.465 

0.445 

0.430 

600 

0.735 

0.685 

0.655 

800 

1.045 

0.945 

0.885 


Respectable results can be achieved by such cali¬ 
brations in spite of the averaging and fitting to 
K/i 2 . For example, for caliber 0.50 API M8 am¬ 
munition the maximum error in the plane of action 
is of the order of 7 milliradians. 83 (This is not the 
total Class B error.) 

5.6.3 Effect of Target Course 

Curvature 

A fighter under attack by another fighter at some 
small angle off tail will frequently attempt to in¬ 
crease the attacker’s deflection and make his aiming 
more difficult by banking as steeply as possible to 
cross the attacker’s bow. (The quarry may also 
dive or attempt to utilize his propeller wash.) For 
attacks in the rear hemisphere, a longer time of 
flight is needed against the circular target path than 
is required by the rectilinear path. The reason is 
that the tracking rate a is the same at a given instant 
for the curved path and for the tangent to that path. 
But the incurving of the target calls for a greater lead. 
Hence t m must increase. This effect can be demon¬ 
strated analytically. 41a The increase required of t m 
can be 41a> 187 from 10 to 20 per cent. But the error 
induced by using a t m appropriate to rectilinear paths 
against curved paths rarely exceeds 6 milliradians 
for expected conditions. 

5.6.4 Irrelevance of Angle of 
Attack, Skid, Slip, Offset Guns 

It will be recalled from Section 3.3.1 that a fighter 
pilot aiming by eye must make a slight aiming allow¬ 
ance along the median line of his sight to account for 


CONFIDENTIAL 












80 


LEAD COMPUTING SIGHTS 


the angle of attack of his guns. The gyro sight in¬ 
cludes the computation of this effect. 84 In fact, a 
gyro sight will give the kinematic deflection regard¬ 
less of skid, slip, or guns deliberately offset at any 
angle, as long as a path is flown as dictated by per¬ 
fect tracking and ranging. This, in itself, is a major 
improvement over eye shooting which must be done 
from a cleanly flown aircraft. No demonstration of 
the point is needed since a fighter in such a situation 
can be thought of as a special case of a turret in a 
bomber. 

5.7 OTHER SIGHTS 

5.7.x The K-8 Sight 

The K-8 electrical sight was used to a limited ex¬ 
tent in the Martin upper turret of the B-24 during 
World War II. It is also of the time-of-flight — 
angular-rate type and, like the K-3 class, measures 



rates relative to the gun platform instead of relative 
to the air mass. However, instead of using a me¬ 
chanical circuit, it computes electrically, generating 
appropriate voltages, utilizing attenuator circuits, 
and employing ultimately its computed voltage out¬ 
puts to displace its sight line laterally and vertically 
by motors. In general, it will behave like the K-3 
in that false rates will be generated when the bomber 
yaws, pitches, or rolls, and in that its decomposition 
of deflection leads to rather large errors at high eleva¬ 


tions of fire. From the point of view of inputs and 
operation it is much like the K-12. A switch permits 
the time-of-flight calibration to be reduced by 10 
per cent to counter, in a blanket fashion, pursuit 
curve attacks. The mechanism will not be con¬ 
sidered in detail, but it is worth while knowing, for 
possible applications, that such techniques have been 
worked out. 

5.7.2 Other Methods of Gyro 

Control 

Gyroscopes may be used in fire control in combi¬ 
nation with forces other than eddy currents. Two 
different schemes 218a are shown in Figure 22 on one 
turntable. The mechanisms should be self-explana¬ 
tory and are set up for performance in azimuth only. 

5.7.3 German Gyro Sights 

The major German gyroscope sights were the 
EZ40, EZ41, EZ42, and EZ45. The EZ40 is a single 
gyro sight with a mirror on the gyroscope to deflect 
the line of sight. (The ray from the reticle hits this 
mirror perpendicularly in the neutral position so 
that there will be no optical dip.) The EZ41 is a 
single gyro sight with the gyro unit remote from the 
sight head. The optical system is therefore actu¬ 
ated by motors controlled by a follow-up system. 
The EZ42 is a twin gyro sight installed in fighter 
aircraft (FW190, Me262). The two gyros are placed 
aft, one being mounted with its axis parallel to the 
longitudinal axis of the aircraft and the other with 
its axis parallel to the vertical axis. Friction dash- 
pots are provided for smoothing and the gyros are 
further constrained by springs. Gyro deflection is 
picked off by tiny potentiometers and the lead is 
computed electrically. A servo drive, employing a 
very small two-phase motor, supplies motion to the 
optical system. (The Zeiss technique of winding 
variable potentiometers and certain other com¬ 
ponents of this sight are well worth keeping in mind.) 
The EZ45 is a remote single gyro sight. The gyro is 
controlled by erecting coils and its supporting case 
is driven by a small motor. Through a time-of-flight 
potentiometer the angle through which the case 
turns is made proportional to the lead. The sight 
parameter is 0.33. The position of the case is 
transmitted by servo to the sight head. A polarized 
relay of high sensitivity is used. 138 


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SUMMARY 


81 


5.8 SUMMARY 

The introduction, Section 5.1, defines lead com¬ 
puting sights as devices which multiply the angular 
rate of the line joining gun to target by an appropri¬ 
ate time-of-flight multiplier to get an estimate of 
kinematic deflection. Ballistic deflection is separately 
obtained and combined with the kinematic deflection. 
The sights considered neglect the rate of change of 
range (and so cannot estimate future range well) but 
may have a suitably chosen time-of-flight multiplier 
to ameliorate this situation. The gunner has only 
indirect control over the line of sight since he controls 
only the guns, from which the line of sight is dis¬ 
placed by the mechanism. 

The digression of Section 5.2 dismisses briefly early 
eye estimation methods based on the rate-time prin¬ 
ciple. The Elephant , ABC , and Apparent Speed 
methods attempt to make, in a mild way, a lead 
computing sight out of the gunner himself. 

Section 5.3 considers the underlying theory com¬ 
mon to all types of lead computing sights. Due to 
tracking errors the rate input must be smoothed 
before being used. The basic equation of these sights 
is discussed through the following points (1) the 
interpretation of smoothing as the process of taking 
an exponentially weighted average of all past values 
of the tracking rate, (2) the damping of oscillatory 
tracking errors, (3) the exponential decay of false 
leads (and slewing routines designed to minimize the 
decay time), (4) the amplification in going from os¬ 
cillatory sight pointing errors to gun pointing errors, 
(5) the meaning of operational stability, (6) the de¬ 
lay in the sight’s presentation of lead, (7) the factors 
affecting the choice of the sight’s smoothing param¬ 
eter, and (8) the analogue between aided tracking 


and lead computing sights. The section concludes 
with a discussion of time-of-flight calibration and 
the Class B errors of sights in general. 

Section 5.4 discusses the mechanical sights of the 
Sperry series in some detail. The ball-cage and cam 
circuit that solves the problem is built up from first 
principles. By writing out the complete sight equa¬ 
tions (consisting of those for the kinematic deflection 
components, those of the blueprint ballistics, and 
those governing the motion of the optical system) and 
the solution of that system, the causes of Class B 
errors are segregated into such compartments as: 
neglect of curvature, gun roll, feedback, and inter¬ 
change of ballistics. 

Section 5.5 is concerned with single-gyroscopic 
eddy current sights. After explaining how a gyro 
may be used to measure kinematic deflection, the 
schematic details of the electromagnetic system, the 
optical system, and the electric circuits are given. 
The variation in the sight’s parameter and the phe¬ 
nomenon of optical dip are explained. Finally, the 
component causes of Class B errors (calibration, 
optical dip, electromagnetic dip, etc.) are reviewed 
in some detail. 

Section 5.6 points out that only sights of the gyro¬ 
scopic principle can be used in fighters. The cali¬ 
bration of time of flight is more effective than that for 
a bomber’s sight since there is only one direction of 
fire and since ballistics may be neglected. This 
calibration is given. The section considers briefly the 
problem of fighter versus fighter (curved target path) 
and the reliability of kinematic lead computation 
when the fighter’s guns are not pointed in the direc¬ 
tion of flight. 

Section 5.7 supplies brief descriptions of the K-8 
electrical sight and various German gyroscopic sights. 


CONFIDENTIAL 



Chapter 6 


CENTRAL STATION FIRE CONTROL 


6.1 INTRODUCTION 

6.1.1 Advantages and Disadvantages 

of Remote Control 

C entral station fire control [CFC] is an arma¬ 
ment system in which a gunner located at a 
sighting station in one part of an aircraft can aim 
and fire guns located in a turret at some other part 
of that aircraft. For the B-29 airplane, for example, 
there are two 2 X 0.50 upper turrets, two 2 X 0.50 
lower turrets, a and one 2 X 0.50, 1 X 20-mm tail 
turret. There are also five sighting stations: a 
pedestal station in the nose, a ringsight at middle 
top, two pedestal side-blister stations, and a tail 
pedestal station. The control of the turrets by the 
sighting stations is flexible — an important feature 
of remote gunnery. The two upper turrets are con¬ 
trolled by the upper station. The forward lower 
turret is under primary nose control and secondary 
blister control. The rear lower is also controlled by 
the two blisters. The tail turret is under primary tail 
control and secondary blister control. Primary con¬ 
trol must be released before secondary control is 
operative. 

Remote control has many advantages. Among 
these may be listed (1) guns and gunners may be 
located in the most effective positions for coverage 
and for vision, (2) loss of a station does not mean 
loss of the guns, (3) the turrets may be smaller and so 
afford less drag on the airplane, (4) pressurization of 
the airplane is simplified, and (5) the firing of the 
guns does not disturb the tracking. 

On the other hand there are equally real disad- 


a The system is committed to upper-lower hemisphere 
coverage. If an attacking plane changes hemispheres the 
upper guns must cease fire and the lower guns take over. 
Should the lower guns be pointed to the wrong side, as they 
might well be when the airplane is under coordinated attack, 
the time delay in bringing them to bear in correct deflection 
position may be serious. 


vantages. Among these are (1) an additional cor¬ 
rection for parallax due to the material displacement 
of sight line and gun line must be made, (2) torsion 
and bending of the aircraft structure may drastically 
impair harmonization, (3) relatively heavy and 
complex follow-up systems of control are needed, b 
and (4) the delays of the follow-up systems may be 
serious. 

6.1.2 Elements of Central Fire 
Control System 

The basic elements of the central station fire con¬ 
trol system are: a sighting station, a follow-up sys¬ 
tem, a turret moved by the follow-up as dictated by 
the sighting station, and a computer connecting the 
sighting station and turret. The computer causes 
the gun line to differ from the sight line by an angle 
just large enough to compensate for the relative mo¬ 
tion of the target during the time of flight of the 
bullet, to compensate for the divergence of the bullet 
from a straight line attributed to ballistic effects, 
and to compensate for parallax. 

The computer receives continuous inputs of pres¬ 
ent range and lateral and vertical tracking rates 
from the gunner who, by keeping the target spanned, 
sets a potentiometer properly, and by keeping the 
target centered, causes two gyroscopes to set up 
precessional torques. The computer also receives 
the azimuth and elevation of the sight line auto¬ 
matically by selsyns at the sighting station; and, 
finally, receives altitude, temperature, and airspeed 
from a hand set operated by the navigator. Parallax 
and the basic ballistics of the ammunition used are 
built into the computer. 


b But, neglecting follow-up equipment, the B-29 armament 
installation weighs 3,316 lb; whereas an installation con¬ 
sisting of two Martin upper turrets, two Sperry ball turrets, 
and one Consolidated tail turret, plus five K-8 sights would 
weigh 4,124 lb. 


82 


CONFIDENTIAL 




BASIC THEORY 


83 


6.1.3 Restriction of Discussion to 
2CH Computer 

In this chapter the nature of the type 2CH com¬ 
puter used in this system will be studied in detail. 
The treatment is theoretical and schematic rather 
than mechanical. The important question of overall 
performance of the gunner-target system is not dis¬ 
cussed. This is properly a matter of elaborate ex¬ 
perimentation both in the air and in the laboratory. 
Nor are considerations in the large such as the design 
of formations for maximum and most efficient fire¬ 
power coverage of the attack region around the 
group, and the support fire problem considered. 

6.2 BASIC THEORY 

6.2.1 Trajectory Equations in 

Vector Form 

The type 2CH computer (General Electric) as¬ 
sumes that the target moves in a straight line at 
constant speed during the time of flight of the pro¬ 
jectile. It is necessary to recast the deflection theory 
of Chapter 2 for this situation to explain adequately 
the functioning of the computer. The methods o 
vector algebra are most appropriate. 35 f 



Figure 1 shows the Siacci range (distance covered 
in the air mass), R, and the corresponding Siacci 
coordinates, P (measured along the line of departure 
to a point directly above the projectile), and Q 
(measured in a vertical plane). 0 It also shows the 
two future ranges peculiar to remote gunnery; D G 

c Unit vectors carry the subscript 1. 


(the distance between gun and projectile), and D 0 
(the distance between the observer-gunner and 
projectile). 

The fundamental equation used in designing the 
computer may be obtained readily from the figure. 
Evident triangular relations are 

P — R + Q> = D 0 + A, Uo = Vo + V(7, 

Dg = R — tfVG, 

where t f is the time of flight, and u 0 is the speed of 
departure P Q . Note that these are solely trajectory 
considerations. The presence or absence of a target 
is irrelevant. Since 

u 0 = ^P = ^(R + Q ), 

the above relations combine to yield 

Vo = 5 Do + P A + P Q+ZVg ’ (1) 

where 


6.2.2 Angular Ballistic and 

Parallax Corrections 

It is necessary to put ballistics and parallax clearly 
in view. To do so, introduce two vectors, B and P*, 
each perpendicular to the muzzle velocity v 0 . The 
ballistic vector B is directed from the bore axis to the 
projectile. It includes both trail and gravity drop 
and subtends at the gun the total ballistic deflection 
angle A b . (Minor effects such as windage jump and 
drift are neglected.) The parallax vector P* is 
obtained, in the first instance, as the perpendicular 
from the sighting station to the bore axis. The dis¬ 
tance from the gun along the bore axis to the point 
of origin of B is CiV 0 where C\ is to be determined. 
Similarly the distance from the gun along the bore 
axis to the point of termination of P* is c 2 v 0 . Conse¬ 
quently 

Do = P* + (d - <*)v 0 + B. (2) 

If, now, the vector (ci — C 2 )v 0 is moved parallel to 
itself so as to emanate from the sighting station, then 
P* may be moved to join (ci — C 2 )v 0 to B. This is a 
second interpretation of P* under which it sub¬ 
tends at the gun the parallax angle A p . The angles 
Ab and A p are called the angular ballistic correction and 
the angular parallax correction respectively. The 


CONFIDENTIAL 






84 


CENTRAL STATION FIRE CONTROL 


next task is to analyze the structure of B and P*. 
Componentially 

B = c 3 v 0 + C 4 VG — Q, 

P* = C2V0 - A. 

To determine the four quantities Ci, C2, c 3 , c 4 , two 
conditions are obtained by taking the dot product 
of each of the preceding two relations by v 0 . The 
third and fourth conditions arise if B and P* are 
substituted in equation (2) and the resulting co¬ 
efficients compared with those of equation (1). One 
finds that 

P*(, lVG \ Q „ 

Cl = — 11 — — cos y G ) — - cos Z , 

U 0 \ Vo / Vo 

A 

c 2 — COS y A , 

Vo 

Q 7 , lv G P * 

C 3 = — COS Z + -COS 7g j 

Vo VoUq 



Uo 


where 7 g is the angle between bore axis and platf orm 
velocity, 7 a is the angle between the vector A and 
the bore axis, and Z is the zenith angle (the angle 
between the vertical Q and the bore axis). 

It is convenient for the sequel to reduce equation 
(2) to the scale of v 0 i (a unit vector in the direction of 
v 0 ). Division of equation (2) by (ci — 02)^0 will yield 
the form 

raDoi = V01 + p + b. ( 3 ) 

(We shall not stop to write out m, p, and b explicitly.) 
The form (3) is adopted since p = tan A p and b = 
tan A*, so that p and b themselves are excellent 
estimates of the two angular corrections. 


6.2.3 Kinematic Deflection 

As a final step in obtaining the basic equations to 
be mechanized, the target and its track are intro¬ 
duced. Since the corrections have been dismissed 
above, only the kinematic deflection need be de¬ 
duced. If v T is the target velocity and r is the 
present range (the range at the instant of fire), then 
relative to the observer 

D 0 = r + (vt — v G )t f = r + rtf, 

which is an exact expansion, since the relative 
target path is straight and the target speed is con¬ 
stant. But r = rri and r = hi + rco x ri, where c0 
is the angular tracking velocity. (This is the radial 


rate rti plus the normal rate rco x ri whose magni¬ 
tude is rco • 1 • sin tt/2.) Using this value for r, we 
find 


where 


w'Dqi = fi + ntfG) x ri = ri + Ak, 


D01 

r + rtf 


(4) 


and 


n = 


r + rtf 


Ak = nt/j) x n. 


(5) 


But co x 1*1 is perpendicular to ri, and also to co since 
co is perpendicular to ri. Hence A * is to be in¬ 
terpreted as the kinematic deflection. Finally, since 
ri x (co x n) = to, the tracking velocity is given by 


co — fi x Ah • 
ntf 


(6) 


6.2.4 Time of Flight 

The time of flight t f is, strictly, a function of D<?, 
v G , Vqi, and the relative ballistic air density p. But 
vg and p can be regarded as parameters, i.e., inputs 
set in before an attack, and present range r is to be 
used instead of future range. Hence, we write 

ntf = F(r , r 0 i). (7) 

Since the vector r 0 i is an argument it is clear that 
direction of fire is to be taken into account by this 
system. 

6.3 TURRET AND SIGHTING STATION 

6.3.1 Follow-up System 

Brief prefatory remarks on the follow-up mech¬ 
anism will supply better orientation for what follows. 
Control consists of (1) a follow-up system, (2) a 
computing unit, and (3) auxiliary circuits. The last 
of these is rapidly dismissed as being of engineering 
interest only. The auxiliary circuits are: starting 
circuits for the rotating machinery, limit-switch cir¬ 
cuits, firing circuits, control-transfer circuits, and the 
like. The follow-up system may be considered in 
slightly greater detail since it indicates that remote 
gunnery does not use the disturbed reticle principle 
of Chapter 5. Suppose that the computer has been 
removed. The operator moves by hand a light 
counter-balanced sight head. The sole job of the 
follow-up system is to make the gun direction v 0 i 


CONFIDENTIAL 






TYPE 2CH COMPUTER 


85 


exactly the same as the sight line direction ri. (In 
a disturbed reticle system the operator controls v 0 i 
which in turn dictates ri. There is nothing in the 
concept of remote gunnery, however, that precludes 
the use of the disturbed reticle system.) 

The follow-up system is schematized in Figure 2. 
The selsyn generator measures sight position by the 
position of its rotor and transmits this position to the 
selsyn control transformer which measures the differ¬ 
ence between gun and sight positions. If a differ¬ 
ence exists the error is sent to a vacuum-tube ampli¬ 
fier which builds up the signal sufficiently to control 
the output of a d-c amplidyne generator. The out¬ 
put of this amplidyne generator controls the d-c 
turret motor which drives the guns into parallelism 



Figure 2. Follow-up system schematic. (Courtesy 
of General Electric Company.) 


with the sight and causes the error signal to vanish. d 
(The peculiar advantages of the amplidyne generator 
are that the output voltage goes to its final value 
very rapidly, and that a very small field current of a 
few milliamperes is required to produce full voltage.) 
The dynamotor is simply a motor-generator set 
which takes dc and supplies ac to the selsyns and 
servoamplifier. In the actual system two sets of 
selsyns are used since a selsyn can be in error by 0.5°. 


d See definition of a servomechanism in footnote to Sec¬ 
tion 8.2.3. 


The one-speed selsyns make one revolution for each 
revolution of the turret and so correct gross errors. 
The 31-speed selsyns make Si revolutions for each 
revolution of the turret and refine the work of the one- 
speed selsyn to yield an error only Vsi of that of the 
one-speed unit. 

6.4 TYPE 2CH COMPUTER 

6.4.1 The Problem for the Computer 

The problem put to the computer is the mechaniza¬ 
tion of equations (3), (4), (5), and (7). In operation, 
it is an interesting combination of gyroscopic proper¬ 
ties, vector-mechanical constructions, and a shifting 
back and forth from electrical to mechanical signals. 
It is not calculating formulas but is effectively solving 
vector equations. The radical difference between 
the 2CH computer and those lead computing sights 
considered in Chapter 5 is that the 2CH is a con¬ 
tinuously correcting system, that is, the answer at 
hand is continuously compared with the answer re¬ 
quired by the inputs, and steps are taken to correct 
the gun position. The net result of the computer’s 
working in conjunction with the follow-up system is 
to displace the guns from the line of sight by a total 
lead angle A, by which account is taken of the kine¬ 
matic deflection, of ballistics, and of parallax. Kine¬ 
matic correction pays attention to the true rate of 
rotation of the gun-target line, since gyroscopes are 
used, i.e., the rates are not relative to the gun mount. 
Ballistics, however, do depend on orientation of the 
guns with respect to the gun mount. 

6.4.2 Rapid Review of Method of 

Solution 

To emphasize the nature of the device as a con¬ 
tinuous comparator, one can start with the total cor¬ 
rection motors in discussing Figure 3. Suppose that 
the rotor has a certain angular displacement cor¬ 
responding to some assumed total lead A. Suppose 
that this deflection is not the correct total lead. The 
output of the correction motor goes to differential 
selsyns which are also supplied with the azimuth and 
elevation of the sight line ri through signals from the 
sighting station selsyn generators. The electrical dis¬ 
placement of the rotors of the differential selsyns 
corresponds to the sum of sight position and total 
lead, and the guns are positioned according to this 
sum ri + A by the follow-up system. 


CONFIDENTIAL 



































86 


CENTRAL STATION FIRE CONTROL 



ALTITUDE 

TEMPERATURE 


ALTITUDE 
V G (IAS) 
TEMPERATURE 


Figure 3. Box schematic, type 2CH computer. 


The differential selsyn signals also go to input units 
which translate electrical values into mechanical 
values and send on present gun position v 0 i to paral¬ 
lax and ballistic units. These units determine the cor¬ 
rections p and b and transfer them to a pantograph 
which actually performs a vector addition of these 
two quantities. Thus one input to the potentiometer 
resolver is p + b. 

Consider the other input to the resolver. The out¬ 
put of the correction motors goes to the axis converter 
which also receives gun position from the input unit. 
The converter combines gun position and total lead 
to produce by mechanical construction the unit 
present range vector ri. The potentiometer resolver 
now performs an operation equivalent to eliminat¬ 
ing D 0 i between the equations 

mD 0 i = v 0 i + p + b 

and 

m'Doi = 1*1 -J- A k, 


and produces as its output the assumed kinematic de¬ 
flection A * corresponding to the assumed total lead. 
It was necessary to obtain \ k since this value must 
be compared with the rate-time value from the time 
circuits and gyros. 

The time-of-flight circuit, which has present range 
(and certain parameters) as input, has a total electric 
resistance ntf. Consequently the current flowing in. 
the circuit is (1 /nt f ) A*, and it is this current that goes 
to the gyroscope unit. In that unit, the current pro¬ 
duces through electromagnets a torque which opposes 
the precessional torque on the gyro generated by the 
tracking. The total torque corresponds to the differ¬ 
ence between co x ri and (l/nt/)\ k . Depending on the 
sign of the difference, contacts in the gyro unit are 
made and one or the other of two relays trips. The 
relays control the total correction motors which 
then back up or go ahead to give an adjusted value 
of A. And the cycle is repeated, or, rather, everything 
is happening simultaneously. 


CONFIDENTIAL 





























TYPE 2CH COMPUTER 


87 


The preceding presentation used vectors as inputs 
and outputs. Except for the parallax and ballistic 
units and the pantograph, this is not quite the truth. 
The correction motors have azimuth and elevation 
components of the total lead as output since these are 
to be used to position the turret in azimuth and ele¬ 
vation. Again, the angular rates measured by the 
two gyros at a sighting station are vertical and lateral 
rates. That is, if the sight head is elevated, the 
“lateral” gyro is measuring the rate of rotation about 
an axis lying in a vertical plane and perpendicular to 
the line of sight while the “vertical” gyro is measur¬ 
ing the rate of rotation about an axis lying in the 
horizontal plane and perpendicular to the line / of 
sight. Thus the outputs of the potentiometer re¬ 
solver are in reality the lateral and vertical com¬ 
ponents of kinematic deflection. 

6.4.3 Behavior of Specific Units 

This subsection will discuss in more detail the in¬ 
ternal nature of the component parts of the 2CH 
computer. Since the approach will emphasize the 
underlying theory, it is complementary to manuals of 
engineering detail. 208 

Parallax Unit 

This is a component in which there is effectively 
constructed in miniature a vector replica of some 
part of the environment. In forming such a replica 



there is no need for duplicating orientation — it is 
relative position that is to the point. Choose, then, 
the vertical direction in the computer to represent 
the gun direction v 0 i. The axes of lateral and vertical 
gun rotation determine a horizontal plane in the 
computer in consequence of this choice. Only the pro¬ 
jection of the vector A (which gives the distance be¬ 


tween sighting station and gun emplacement) on the 
longitudinal axis of the fuselage will be used. This 
length is a fixed physical distance along a lever in the 
unit and is factory set. In order to get p we must first 
obtain the projection of A on the horizontal plane of 
the parallax unit and then modify this length ac¬ 
cording to the range. Schematically, these steps are 
accomplished in Figure 4. The input unit translates 
electric signals into gear rotations to position A as 
shown. The projection on the horizontal plane is pre¬ 
cisely the scaled length of a line segment from the 
sighting station dropped on the bore axis, i.e., P* in 
Figure 1. This motion is reflected by sliding rods as 
corresponding motion of the point a (Figure 4). If 
the fulcrum / on a vertical lever (waggle-stick) is 
moved up and down as a function of range then the 
point b will be displaced by the angular correction p. 

Ballistic Unit 

This component is very similar to the parallax 
unit. It differs in that the length B that plays the role 
of A must vary with altitude, IAS , and temperature. 
This is accomplished by gears actuated by the navi¬ 
gator’s handset. Although the mechanization differs, 
the result in the plane of the pantograph is a motion 
of one end of another waggle-stick which represents b 
and lies in the plane of gun lateral and vertical axes 
just as does p. 

Pantograph 

Since p and b are now in the same plane, the pan¬ 
tograph can add them vectorially very readily. The 
device is a jointed framework of metal strips schema- 


8 



Figure 5. Pantograph unit, schematic. 


tized in Figure 5. The parallax waggle-stick drives F 
and the ballistic waggle-stick drives A. Under a dis¬ 
placement from the original position (A 0 ,F 0 ,C o) of 
amounts A 0 A = — b and F 0 F = (^)p, if is readily 
found from the geometry of the frame that CqC = 
p + b. 


CONFIDENTIAL 







88 


CENTRAL STATION FIRE CONTROL 


Axis Converter 37 

Conceptually, this assemblage of gears and shafts 
has as inputs the azimuth and elevation components 
A a and A E of the assumed total lead A and the gun 
elevation Eg . (Actually, the inputs are three gear 
rotations of amounts 2A E — 2Aa — Eg , Eg + 2Aa, 
and E g respectively.) As in the preceding units the 
vertical direction in the computer is selected to 
represent the gun direction v 0 i. The horizontal plane 
in the computer is, then, the equivalent of the spatial 
plane determined by gun lateral and vertical axes. 
This system of three directions — gun direction and 
the two axes — is fixed in the unit. The job of the 
axis converter is to construct in correct relative posi¬ 
tion to this fixed system the corresponding system of 
the sight line, i.e., sight direction ri, and sight lateral 
and vertical axes. This it does by performing two 
rotations of the physical length corresponding to the 
vertical gun direction. Since the situation in the 
computer is only a reoriented version of the spatial 
picture we may revert to space for clarity. In space, 
rotation of a line in azimuth is represented by a 



vector wi directed vertically upward. Rotation in 
elevation is represented by a vector ta E lying in the 
horizontal plane and perpendicular to the vertical 
plane through the line in question. Lateral rotation is 
represented by a vector w l lying in the vertical plane- 
through the line and perpendicular to that line. 
Vertical rotation is identical with elevation rotation 
so that C Ov = 03 E . The angle between col and c ca is 
clearly the elevation E of the rotating line. Hence 
03 l = o )a cos E. Figure 6 illustrates the situation, 
(co = o )a + o) E is the total angular velocity, and when 


the decomposition co = co^ + WL + W(?is made, 03 G is 
an additional and necessary angular velocity — the 
gun roll.) It follows from these rate relations, o)v = o) E , 
o)l = o )a cos E, that, correct to first order terms, e the 
sight position can be obtained from the gun position 
by a vertical rotation of amount — A E followed by a 
lateral rotation of amount —Aa cos Eg. This is done 
by gears in the axis converter. A representation of ri 
thus appears as a mechanical displacement in a hori¬ 
zontal plane in the potentiometer resolver. 

Potentiometer Resolver 37 

It will be recalled that the function of the resolver 
is to take in the sight direction and the sum of the 
parallax and ballistic corrections as two vector-me¬ 
chanical inputs, and to give out voltages Akv and A kL 
which are the vertical and lateral components of the 
assumed kinematic deflection. As expected within 
the resolver is a vertical direction representing gun 
direction v 0 i. The plane of the pantograph is per¬ 
pendicular to this vertical direction and the output 
of the pantograph gives the displacement C 0 C = 
p + b. Hence in the resolver OC = OC 0 + CoC= 
V01 + p + b is constructed. According to equation 
(3) the device has physically produced the vector 
raDoi. Another part of the resolver is rigidly affixed to 
the axis converter. In Figure 7, this part appears as 


PANTOGRAPH PLANE 



Figure 7. Potentiometer resolver, schematic. 


OS'. At the point S' the resolver has a plane perpen¬ 
dicular to OS'. A waggle-stick from the point C in the 
pantograph plane is maintained so that C'C is parallel 
to OS'. The vector equation OC' = OS' + S'C' is 
translated as 

_ w'Doi = ri + A*. 

e The mechanism actually reproduces the geometrical 
constraints of the situation and so is not guilty of approxima¬ 
tions. 


CONFIDENTIAL 










TYPE 2CH COMPUTER 


89 


By this frontal approach to the problem, i.e., a 
construction in miniature of the vector situation in 
space, the kinematic deflection has been produced. 

It remains to obtain the lateral and vertical com¬ 
ponents of the kinematic deflection relative to the 
sight line in the form of voltages. In the resolver 
plane, which is perpendicular to the computer’s re¬ 
production of the sight line, S'H and S'M are such 
components. A voltage equal in magnitude to A kL 
can be obtained by means of a variable potentiometer 
whose sliding contact M moves along the line S'M 
in the lateral direction. And with some mechanical 
elaboration the same remarks apply to A k v. 

Time-of-Fljght Circuits 

The circuit is a combination of fixed and variable 
resistances such that the total resistance approxi¬ 
mates ntf. The schematic is shown in Figure 8. The 



j-- n, t-* 

Figure 8. Time-of-flight circuit, schematic. 

variable range resistance, controlled by the gunner 
who keeps the target framed, supplies a continuous 
estimate of present range. The navigator’s hand set 
puts in altitude, indicated airspeed and temperature. 
The ballistic unit supplies the component of own 
speed in the direction of fire, which means that a 
good value for velocity of departure is at hand. 
Certainly all the appropriate variables are present 
to obtain an excellent value for time of flight. How¬ 
ever, in one direction it is a question of how well a 
simple circuit can approximate a complicated func¬ 
tion, and in another direction what calibrations can 
be made to allow for curved courses. It should be re¬ 
membered that for straight tracks the results with 
respect to this second direction should be excellent 
since nt f is exact by equation (4). 


6.4.4 The Gyroscope Units 

The gyroscope units 36 will be discussed in some¬ 
what more detail than w^re the preceding com¬ 
ponents. To measure lateral and vertical tracking 
rates there are twp gyroscopes at each sighting sta¬ 
tion. Each gyroscope consists of a flywheel and the 
rotor of a small, high-speed, and constant-speed 
electric motor running on 27 volts d-c at 10,000 revo¬ 
lutions per minute. The lateral rate unit is shown in 
Figure 9. The housing axis is always parallel to the 



Figure 9. Gyroscope, type 2CH computer. (Courtesy 
of General Electric Company.) 


line of sight, and the axis of the pins that support 
the gyro motor frame remains horizontal. The gyro 
spin axis may depart very slightly from parallelism 
with the sight line, but this departure is in elevation 
only and is limited by adjustable stops. Conse¬ 
quently, it will not be inferred that the gyro axis 
displacement measures deflection as it did in Sec¬ 
tion 5.5.2. 

Under rotation about the lateral axis (this is w L ) in 
Figure 6, the gyro exerts a precessional torque about 
an axis perpendicular to both the spin axis and the 
axis of the applied tracking torque and hence about 
the axis of the supporting pins. But bucking this 
precessional torque is an electromagnetic coupling 
due to the displacement of permanent magnets 
(which are fixed to the gyro frame) from the center 
of an erecting coil fixed to the housing. The current 
in the erecting coil is A kL/ntf, and the torque pro¬ 
duced is proportional to this current. If this torque is 
not as strong as that due to precession the gyro will 
precess slightly until an electric contact is met. 
For this case the assumed A kL was too small. Hence 
the contact causes a relay to trip which permits a 
current to flow in the armature of the total correction 
motor in a direction which will increase A k L. This will 
strengthen the erecting coil torque and pull the gyro 
away from the contact. If the erecting coil torque 


CONFIDENTIAL 


















90 


CENTRAL STATION FIRE CONTROL 


had been too strong initially the process would have 
reversed. In general, the gyro will hunt or successively 
under-and-over correct by going back and forth be¬ 
tween the contacts. Back lash is deliberately intro¬ 
duced between the total correction motor and its 
output wheel to minimize hunting. Only under per¬ 
fect and uniform conditions will the gyro ride with 
its spin axis identical with the housing axis. 

It will be noticed that there is no smoothing in the 
precise sense of that given under “Weighted Averages” 
of Section 5.3.4. However, the gyro housing is at¬ 
tached to the sight by a form of elastic coupling 
damped by friction. The basic purpose is to reduce 
vibration but possibly a certain amount of smooth¬ 
ing is also introduced by such a support. 

The gyroscope system introduces certain errors in 
the problem which can be understood by writing out 
the dynamical equations of the system. 36 Let Mg and 
M P denote, respectively, the vector angular mo¬ 
menta of gyroscope and gyro motor frame. Let — T G 
be the moment of all forces exerted by the gyroscope 
on the frame. (This includes reaction and bearing 
torques of the motor.) And let T H be the moment of 
all forces exerted by the gyro housing on the gyro 
motor frame. (This includes the erecting coil torque, 
frictional torques, pivot reactions, and reactions 
when the stops are closed.) The equation for the 
gyro motor frame is 



Gy 


G, 


and that for the gyro is 
dM G 
dt 

so that 

d( M/? -f- Mg) 
dt 


= T 


H , 


in a fixed coordinate system. With respect to a ro¬ 
tating coordinate system determined by the pin 
axis, the spin axis, and an axis perpendicular to 
these/ and with d/dt now measuring rates of change 
with respect to this rotating system, a theorem from 
mechanics yields 


d{ M f + Mg) 
dt 


+ 12 x 


(Mf + Mg) = T#, 


where 12 is the vector angular velocity of the ( p,s,q ) 
coordinate system. With an obvious notation for 


1 Components in these three directions will be denoted by 
p, s, and q respectively. 


moments of inertia, the p component of this equa¬ 
tion is 


(iGp + IFp) ~7~ + (JGp IFq 

at 


- I Gs — / f «) 12 9 12 8 

i Gs n q n o = t H p , (8) 


where 12 0 is gyro speed with respect to its frame. The 
purpose of the gyro unit can now be phrased sym¬ 
bolically. It is designed to approximate 12 q (which is 
the tracking rate) by use of the equation 

— 7gs12 ? 12 0 = T Hp , (9) 

where T Hp is essentially the torque exerted by the 
electromagnetic coupling on the spin axis. In other 
words, the device is neglecting the first two terms of 
the complete equation (8). 

Consider the second term on the left of equation 
(8) and assume that the expression in parentheses is 
of the same order of magnitude as T Gs . The angular 
rate 12 s is precisely the equivalent of the gun-roll rate 
of Figure 6, i.e. 12 s = coa sin E, where A and E are the 
azimuth and elevation of the spin axis (which di¬ 
verges very slightly from the sight axis). But c oa 
cannot exceed v sec E/r where v is the velocity of the 
target relative to the bomber and r is the range. 
Hence 12 s is large only if E is close to 90° and/or r is 
small. If | E | < 77° 45', v < 1000 fps, and r > 750 
ft, then | 12 s | is 0.6 per cent of 12 0 (which is 20,000x 
radians). The second term may be safely neglected. 
However this does indicate that certain restrictions 
on high elevation fire are inherent in the system. 

Turning to the first term on the left of equation 
(8), we are concerned with the angular acceleration 
dSlp/dt about the pin axis. The lead given by equa¬ 
tion (8) is 




Tnptf 
I Gs$ 2 q 


+ 


Igp + I Fp dSlp 
IgsV 0 ~di f ’ 


when the second term is neglected. Hence angular 
acceleration introduces a spurious lead (the second 
term on the right of the preceding equation). For 
perfect tracking of an accelerated target this can 
hardly exceed 8 milliradians. 36 But under normal 
imperfect tracking the operator may be making very 
large oscillatory errors in angular position. To what 
extent the computer filters acceleration errors is not 
definitely known. The obvious procedure is to make 
a laboratory bench test. 


6 . 4.5 Total Correction Motors 

The total correction motors make lead corrections 
at a constant rate of about 40 milliradians per 


CONFIDENTIAL 








ANGLES AND ANGULAR RATES IN MILRADS AND MILRADS PER SECOND 


TYPE 2CH COMPUTER 


91 



0 .4 .8 1.2 1.6 2.0 2.4 2.8 

TIME IN SECONDS 

Figure 10. Typical frontal attack. 


CONFIDENTIAL 


RANGE IN YARDS 



































92 


CENTRAL STATION FIRE CONTROL 


second. If the motor runs continuously in the same 
direction for 1.5 seconds, a higher speed of about 80 
milliradians per second becomes operative. To illus¬ 
trate the effect of the time delay caused by this motor 
suppose that with an initial lead of zero the gun is to 
be slewed in one-half second to get on and then to 
track a target requiring a lead of 100 milliradians. 
Then, 85 approximately: at t = 0, A = —20 (sight on, 
tracking started); at t = 0.5, A = — 20 (backlash 
overcome); at t = 1, A = 0; at t = 1.5, A = 20 
(high speed begins); at t = 2.0, A = 60; at t = 2.5, 
A = 100. 

What seems needed is a system of variable speeds 
so arranged that when (and only when) the required 
lead correction is large a high speed will be provided. 86 
This may be accomplished by using the variable pres¬ 
sure of the gyroscope on the stops to introduce vari¬ 
able resistance into the motor circuit. Dynamic brak¬ 
ing may also be used. It should be observed that al¬ 
though increasing the motor speed decreases the time 
delay, at the same time smoothness is sacrificed and 
excessive hunting may occur. 

To illustrate the need of high speed in getting on 
target quickly and of low speed still sufficiently 
great to keep up with lead changes, curves for a 
typical frontal attack are shown in Figure 10. 

In addition to the variable speed proposal noted 
above, various other methods have been suggested. 87 
Typical of these are (1) the motors run at high speed 
except when the trigger is pressed, (2) the shift from 
high to low speed when the trigger is depressed is 
delayed for, say, one second. It is probably worth¬ 
while to make a rough paper assessment of expected 
results in assumed tactical situations under such 
proposals. The whole slewing operation should also 
receive careful attention. 88 

For purposes of reference, the motor equation may 
be quoted. 89 ’ 90 For a d-c motor with separately ex¬ 
cited or permanent field 



a E 

+ R W = R~ I{ - W) ’ 


where a and b are constants, R is the total resistance 
of the armature, brushes, and commutators; E is the 
applied voltage; and I(w) is the current required to 
run the motor at the speed w, if w were constant. 
For frictional loading 

I ( w ) = 7 0 + hw 

is an adequate form, where / 0 depends on load, bear¬ 
ing friction, and armature hysteresis, and hw de¬ 


pends on the armature drag due to eddy currents and 
on viscous friction and is nearly independent of the 
load. 


6.4.6 Range Follow-up Motor 

Another inherent weakness in this computer sys¬ 
tem is that the range follow-up motor is frequently 
too slow. Three ranges must be considered. These are 
(1) the true range, (2) the range introduced by po¬ 
tentiometer at the sighting station, and (3) the range 
set into the computer by the range follow-up motor. 
In its steady state the motor alters computer range 
r (in yards) according to 91 

r = -110 + (r 0 + 110)e -o ' ool216A:< 

where k is the number of revolutions per minute of 
the motor’s output gear. The implications of this be¬ 
havior can be made out by referring to Figure 10. 
Suppose at t = 0 that the true range r 0 = 1,100 yd 
is in both the sight and in the computer. The oper¬ 
ator, behaving perfectly, is changing range at an 
almost constant rate of 320 yd per sec. The maximum 
rate of change of computer range, however, occurs 
for k = 250, which is the maximum motor speed. 
It is 

r = — 0.304(r + 110). 

Hence below 942 yd range the motor cannot run fast 
enough to keep up with the input. The gunner could 
suddenly reduce range to 250 yd and computer re¬ 
sults would be just the same as under perfect rang¬ 
ing. 


6.4.7 Calibration 

In connection with the above remarks on the total 
correction motors and on the range follow-up motor, 
the effect on calibration of the time of flight may be 
noted. In any machine, whenever defects, unavoid¬ 
able or not, call for extensive redesign or installation 
of new units, one attempts to save time and equip¬ 
ment and still achieve respectable overall results by 
giving artificial values to adjustable parts of the 
existing equipment. For example, to account for the 
delay in the solution time on nose attacks, the time 
of flight is multiplied 87 by 1.07 since leads increase 
on nose attacks. But calibration usually has its price. 
In this case performance against attacks from the 
rear will deteriorate since the factor should be less 
than unity in that hemisphere. 


CONFIDENTIAL 



SUMMARY 


93 


6.5 SUMMARY 

Section 6.1 limits the discussion of this chapter to 
a theoretical study of central station fire control it¬ 
self. The wider questions of formations, support fire, 
and experimental results are not considered. 

Section 6.2 derives four basic equations which the 
computer is to mechanize. Significantly, the first 
three are vector equations and in large part the com¬ 
puter considered is a vector mechanism. 

Section 6.3 discusses the nature of a follow-up 
system under which guns remote from an operator 
are made to follow his positioning of the sight line. 


The computer, which introduces the gun displace¬ 
ment from the sight line required by target motion, 
ballistics, and parallax is disconnected. 

Section 6.4 gives first, in schematic terms, the 
method by which the computer solves the total de¬ 
flection problem. Next the behavior of the individual 
components is studied in more intimate detail. Con¬ 
siderable attention is paid to the behavior of the 
gyroscope units since angular tracking accelerations 
can lead to appreciable errors through this system. 
Finally the time delay in response of the total cor¬ 
rection motors, and the inadequate range change rate 
in the range follow-up motor are considered. 


CONFIDENTIAL 



Chapter 7 


ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


7.1 INTRODUCTION 

7.1.1 Role of Airborne Test 

Programs 

The closed feedback loop formed by the successive 
elements — target, gunner, controls, computer, guns, 
bullet, target — is a complicated servomechanical 
system. When a system is too complicated to permit 
an exact theoretical analysis of behavior, the com¬ 
mon technique is to supply certain inputs and com¬ 
pare the actual output with a perfect output. This 
is the philosophy of airborne assessment tests of fire 
control equipment. If it is a question of choice be¬ 
tween two fire control systems, the procedure is 
certainly logical, provided a proper simulation of the 
conditions under which the systems must perform is 
made, and provided that the experimental technique 
is considerably more accurate than the mechanism 
under test. If it is a question of prediction of overall 
performance in combat of a particular system, the 
attitude may still be defended, although not nearly 
as strongly as in the competition situation. But if it 
is a question of the decomposition of overall errors 
into componential causes, so that design improve¬ 
ments may be made, such assessment has minimum 
value, unless it is very complicated and detailed. a 

To meet, in part, this last question, the errors made 
by the system under perfect operation and construc¬ 
tion have been studied earlier in this account. The 
manufacturing variation from instrument to instru¬ 
ment and other variations caused by non-perfect 
construction and functioning can be studied by 
laboratory bench tests in which factors such as turn¬ 
table speeds and temperature can be carefully con¬ 
trolled. More elaborate laboratory machines may 
use perfectly known inputs to a human gunner by 


a Such a detailed assessment program was undertaken for 
the B-29 system by the General Electric Company. 


means of cammed target courses and controlled gun 
platform motion and may measure in complete de¬ 
tail the behavior of each component of the system. 
It is this method that is most fruitful in arriving at 
design changes. 

One cannot expect to add up componential errors 
deduced by the methods of the preceding paragraph 
to obtain the total error determined by airborne 
tests. The reason is twofold. In the first place, the 
complete system is in no sense linear. It has numerous 
feedbacks, lags, and unexpected gunner reactions. 
In the second place, the closed loop described initially 
contains elements which are essentially statistical. 
The main variables are: target behavior (a course 
cannot be reproduced exactly in the air), gunner be¬ 
havior (intra-gunner variations may be as large as 
inter-gunner variations), gun mount motion (which 
varies with the aircraft, the weather, and the pilot, 
particularly under avoiding action), and instru¬ 
mental variation (sights, guns, turrets as affected by 
wear and calibration). 

But it is concluded that field tests, laboratory tests, 
and theoretical analysis all have an important place 
in the field of fire control. The important point is 
that they be coordinated partners in this field. The 
balancing and coordination of their roles is the duty 
of vigorous technical administration. 

7.1.2 Classification of Errors by 
Cause and Statistics 

The errors of the complete system may be classi¬ 
fied systematically. The first decomposition 14>127 
emphasizes cause: 

1. Class A errors. Inter- and intra-instrumental 
variations are caused by play arising from permitted 
tolerances and further play introduced by loose ac¬ 
tion or worn parts. These are errors representing 
the failure of the instrument to operate as designed. 


94 


CONFIDENTIAL 



INTRODUCTION 


95 


2. Class B errors. As defined earlier, these are the 
theoretical errors of the mechanism caused by design 
assumptions and compromises. 

3. Operational errors. These occur because of 
tracking and ranging errors, harmonization errors, 
and faulty presetting of flight data. 

4. Dispersive errors. Because of mount and gun vi¬ 
bration, variation from round to round of the ammu¬ 
nition in ballistics and muzzle velocity, play in a 
turret, and the difficulty of holding a point of aim, 
the bullets of a burst spray out in a cone. Although 
the pattern on a section of this cone is usually ellip¬ 
tical, 154 the orientation of this ellipse so depends on 
the direction of fire and other factors that an equiva¬ 
lent circular pattern is taken conventionally. This 
assumption is augmented by supposing that the 
pattern is described by a circular Gaussian distribu¬ 
tion about a mean point of impact. (See Section 
1.5.2.) 

It should be noted that although the pattern tends 
to conform to normality, the locations of individual 
shots are not independent. This is serial correlation 
and is evidently of most importance in connection 
with very short bursts. Roughly, if one shot has a 
large error of one sign, then the next round is likely 
to have a large error of the same sign. This is related 
to aim wander. 198 As the gunner tries to hold on a 
target, his aim traces a snakelike path. One can 
imagine the small pure dispersion being carried along 
this path as a probability cloud. We attempt to 
make a distinction between the dispersive serial cor¬ 
relation arising when a gunner tries to hold on a 
fixed target and the much more important tracking 
serial correlation. The point has not been clarified 
completely from the theoretical point of view. On the 
whole, we shall attempt to avoid it by using more 
naive statistical ideas. 

Another classification of system errors can be 
made which emphasizes their statistical nature: 128 

1. Fluctuating errors. These errors arise from dis¬ 
persion, tracking errors in angle (including any 
amplification of the system), and Class A errors. 

2. Quasi-steady errors. These give a sensibly con¬ 
stant bias over one burst but vary from burst to 
burst. Typically, they are attributed to range input 
errors, harmonization errors, and faulty setting of 
flight data. 

3. Steady errors. Given a particular situation, such 
as a specified point on an assigned pursuit curve, 
these errors yield a fixed bias. They are evidently 
Class B errors. 


7.1.3 Nature of Modern Airborne 
Test 

Early assessment tests were made by firing 
against a flag target towed past the gun mount 
on a parallel course at low relative speed and close 
range. The number of hits was counted. This 
procedure can be defended only on the basis that 
the installation was airborne and it did fire. Clearly 
an extrapolation of performance on such trivial 
courses to expected performance against an actual 
target course is unjustified. Furthermore, the 
bullets that missed are lost to analysis, since 
their position is not known. Thus the method is 
statistically inefficient. 

Current methods for airborne tests are based on 
the dry run principle. That is, combat conditions are 
simulated as closely as possible by having a fighter 
plane attack the bomber. Neither side fires actual 
bullets. b Instead, by means of various accurate re¬ 
cording instruments, usually airborne by the aircraft 
participating, four basic types of data are obtained 
(1) the range and bearing of the target, (2) the 
tracking and ranging of the sight under test, (3) the 
actual gun pointing, and (4) the attitude, speed, 
and altitude of the gun-mounting aircraft. Some¬ 
times it is desirable to obtain all or some of these 
data, in synchronization, for the target aircraft. 
To these two requirements, combat simulation 
and precise instrumentation, a third must be added. 
For a given type of tactical situation, such as a 
rear quarter attack, sufficient replications of the 
circumstance must be made to give adequate sta¬ 
tistical voice to those variables which determine 
the system’s output but which cannot be con¬ 
trolled. This implies that the experiment must be 
designed carefully, usually by the Latin-square 
technique. In general, a minimum of replications 
of the order of 15 is needed for each system under 
each tactical situation. 

We may now turn to some of the details of assess¬ 
ment programs. The three major phases are (1) in¬ 
strumentation, (2) interpretation and reduction of 
data, and (3) measurement of effectiveness. Only a 
rapid survey is contemplated. 


b The lack of inhibition caused by gunfire is a serious flaw 
of the method which cannot be answered by the use of frangi¬ 
ble bullets, since full-scale combat conditions could not then 
be used. Simulating recoil mechanisms may be much more 
to the point. 


CONFIDENTIAL 




96 


ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


7.2 INSTRUMENTATION (AIRBORNE 
PROGRAMS) 

7.2.1 Bearing and Range of Target 
and Attitude of Gun-Mounting 
Aircraft 

To determine the correct deflection at each instant, 
i.e., perfect output, it is necessary to know the space 
path of the target relative to the gun mount as a 
function of time. This is normally accomplished by a 
determination at each instant of the range and bear¬ 
ing of the target with respect to the gun mount. 
Three 35-mm cameras with lenses of 1-in. focal 
length, when ganged and installed in a waist window 
of a heavy bomber, will cover a field of approximately 
120° in azimuth and 50° in elevation. 177 The align¬ 
ment is determined by photography on the ground. 
The lenses must be calibrated so that the magnifica¬ 
tion will be known under projection. The bearing 
angles relative to this installation can be determined 
through projection on a suitably gridded screen. 
There are other methods of determining bearing. 
The motion in azimuth and elevation of a turret 
which mounts a single camera and which tracks the 
target can be picked off the gear trains either by 
gears or by selsyns. 34 Dial readings are photographed. 
The direction of the center of the picture is known 
and the bearing of the (displaced) target image is 
readily determined. It would also be possible to use 
ultra-narrow-beam radar which can give angles with 
a probable error of 1 or 2 milliradians. Range is 
usually determined by measurements of the image 
on the film supplied by a camera of 2-in. focal length 
(to gain magnification at the expense of field size) 
clamped to the gun. This determination is usually 
the least accurate step in the entire scheme because 
of blurring of wingtips and foreshortening of the 
target. The use of band painted wingtips, of landing 
lights, and of infrared lamps has been suggested. 

To reduce the relative bearings to direction in an 
air mass coordinate system, the yaw, pitch, and roll 
of the airplane must be measured. This is done by a 
pair of suitably gimbaled gyros which maintain the 
direction of their spin axes in space as the aircraft 
rocks about the mounting. The resulting displace¬ 
ments can be picked off either mechanically or by 
selsyns. Yaw, pitch, and roll can also be calculated 
after measurement of the tilt of the horizon and the 
position of a sunspot on some plate fixed in the 
aircraft. 92 


7.2.2 Astrometric Methods 

The space paths of the two aircraft might well be 
determined by astrometric methods. 150 Suppose that 
two vertical cameras of 12-in. focal length are 
mounted on the ground at opposite ends of a suitable 
base line. The aircraft photographed by these 
cameras carries perhaps four brilliant flash units. 
Coincident with each flash a radio signal is sent to 
the ground and recorded by a chronograph with an 
accuracy of 1/1,000 sec. Standard precision methods 
are applied to measure the image positions on the 
two plates. With aircraft at 10,000 ft the overall 
probable error of position is zb 1 ft, and the probable 
error in velocity is ±34 fps. The disadvantages of 
this procedure are found in the restricted sky area 
available to the maneuvering aircraft (this could 
presumably be met by employing a lattice of 
cameras) and in the complexity of the data reduc¬ 
tion. Nevertheless, the flexibility of the installation 
in assessing almost all types of aerial problems should 
make it attractive to a large proving ground. 

7.2.3 Synchronization 

Returning to the standard airborne instruments, 
the gun camera used to determine range also es¬ 
tablishes the actual gun pointing. Finally, if a 
camera is mounted to photograph the combining 
glass of the gunsight, then ranging and tracking 
errors are known; and if a camera photographs an 
instrument panel, flight data are recorded. It is 
evident that these four camera units, tricamera, gun, 
sight, instruments, must be synchronized. This is 
usually effected by placing a light bulb in the back 
of each camera which fogs the film edge in each 
camera at intervals dictated by a master control 
unit. 

7.2.4 Air Mass Coordinate 

Technique 

Somewhat simpler instrumentation is possible if an 
entirely different approach 9S - 137 to the problem of 
determining gun-pointing errors is made. For near 
pursuit curve attacks the bore axes of the bomber’s 
guns and of the fighter’s guns will be close to parallel¬ 
ism, since the leads taken by the duelists are equal 
in size and opposite in sense, to a first order approxi¬ 
mation. (See Section 2.2.2.) Equip each aircraft with 
a camera kept in rigid alignment with the respective 
bore axes. Then the image of the opposing aircraft 


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REDUCTION OF DATA 


97 


may be expected on each film. Suppose, further, that 
the bomber is supplied with a 100 per cent own-speed 
mechanism which places on the film a point giving 
the resultant direction in the air mass that the bullet 
would take under the action of muzzle velocity and 
mount velocity. Suppose that for the fighter the 
angle of skid 94 and angle of attack can be de¬ 
termined so that the resultant instantaneous flight 
direction in the air mass of the fighter can be placed 
on the projected fighter film. 

At a given instant of fire immobilize the bomber. 
Then on the frame corresponding to this instant the 
direction the bullet takes in the air mass appears as 
a point. If the position the fighter would reach in 
the air mass after the time of flight of the bullet 
could be plotted on the same frame, then the gun¬ 
pointing error would be known immediately. To 
do this, suppose that the fighter proceeded over the 
time of flight of the bullet at its instantaneous ve¬ 
locity, i.e., the curvature of its path is neglected. 
The crucial point of the method now appears. The 
plane determined by the fighter’s wingtips and the 
bomber’s camera appears on the bomber’s film as a 
line joining the wingtips of the fighter image. This 
plane appears on the fighter’s film as a line parallel 
to the bottom of the film (since the fighter’s camera 
is so set with respect to the wingtips) and passing 
through the camera installation on the bomber. 
The plane containing the fighter’s instantaneous ve¬ 
locity and the bomber’s camera appears as a straight 
line on the fighter’s film passing through the image 
of the camera installation of the bomber. Since the 
films of the two cameras are nearly parallel, the angle 
between this line on the fighter film and the hori¬ 
zontal on the fighter film may be transferred with 
negligible error to the bomber film with respect to 
the line joining the fighter’s wingtips. This establishes 
a line on the bomber film on which the fighter’s 
future position must lie. But the angle subtended 
at the immobilized bomber by the fighter motion is 
precisely the deflection required by the fighter’s ap¬ 
proach angle (measured on the fighter’s film) and so 
is 

sin a 

qv F - 

Vo 

Consequently, the future target position is estab¬ 
lished on the chosen frame of the immobilized 
bomber’s film record. 

The method now corrects the future position by 
computing the fighter’s normal acceleration. The 


true future position is plotted on a line passing 
through the previous future position and perpen¬ 
dicular to the wingtip line. 

Although the procedure has real advantages from 
the instrumental point of view (no recording of yaw, 
pitch, roll is required), and from the point of view 
of data reduction (ballistics are not used since work 
is in the air mass) its overall errors are only known 
theoretically. It has not been carried through in 
actual programs as has the relative coordinate 
method. 

7.2.5 Distant Reference Point 
Method 

The simplest method of assessment that has been 
devised uses just one camera fixed to move with the 
gun with which it is bore-sighted. 122 It relies on the 
existence of distant reference points such as moun¬ 
tains or clouds which do not move during the assess¬ 
ment either absolutely or relatively. If a certain 
frame is chosen it may be assumed that correctly 
aimed bullets fired at an earlier instant, correspond¬ 
ing to the time of flight for the range of that frame, 
are hitting the target. The film is turned back by the 
appropriate number of frames and the fighter posi¬ 
tion, gun position, and reference points are marked 
on the projection screen. Turning back to the 
original frame, the gun error is known by superim¬ 
posing the reference points, correcting original gun 
position for ballistics and then observing by how 
much the original corrected gun position fails to be on 
the target. 

Despite the lack of accuracy, the simplicity may 
well recommend this method for either rapid work or 
pilot tests designed to show r gross errors. 

7.3 REDUCTION OF DATA 

7.3.1 Techniques 

This section will content itself with a discussion of 
the techniques that have been found useful in re¬ 
ducing the raw T data obtained in airborne test pro¬ 
grams. The computing schedules for particular appli¬ 
cations have been written out in detail 34 > 95 > 113 - 177 and 
are not given here. The techniques are those con¬ 
cerned (1) with the rotations of coordinate systems, 

(2) w T ith ballistic and parallax calculations, and 

(3) with the treatment of single-shot probabilities 
and of measurement errors. 


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ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


7.3.2 Rotation of Coordinates 

It is to be expected that three-dimensional geom¬ 
etry will be most important in assessment programs. 
Examples are (1) reduction of azimuth and elevation 
measured with respect to a coordinate system de¬ 
termined by a tricamera installation to bearings 
with respect to a coordinate system fixed in the air¬ 
craft that mounts those cameras (the cameras are not 
necessarily aligned with the aircraft), 114 (2) a similar 
reduction in going from bearings relative to aircraft 
axes to bearings relative to an air mass coordinate 
system (the aircraft axes are rotated in the air mass 
system because of yaw, pitch, and roll), and (3) a 
transformation of angles measured in a system rela¬ 
tive to a camera fixed in a fighter to angles measured 
in a system at the fighter 45>51 and parallel to those 
at a bomber or at a ground target. 

These examples must deal with the rotation of co¬ 
ordinate systems. The most elegant and painless 
methods of geometry are those employing matrix 
algebra. 34 According to these methods, if x is a vector 
in one coordinate system and x the same vector in 
a rotated coordinate system, the two are connected 
by the matrix equation 


Xl 


CO 

£ 

3 * 


Xi 

X 2 

= 

Vi V 2 Vz 


X 2 

_Xs_ 


JWi W 2 Wz_ 


_^3_ 


or, briefly, 

x = S • x. 

The array S is composed of the direction cosines of 
the axes of the first system with respect to those of 
the second, and matrix multiplication according to 
the usual rule 0 is used. 

The application of the theory is illustrated by the 
problem of the stabilization of coordinates . 34>96 In this 
problem, the azimuth and elevation of a target are 
given in a coordinate system fixed in the bomber. 
The bomber system has been obtained from an air 
mass system by a sequence of three rotations 

(1) a rotation in yaw, with a vertical axis, of angle Y, 

(2) a rotation in pitch, with a horizontal axis, of 
angle P, and (3) a rotation in roll, about the aircraft’s 
longitudinal axis, of angle R. (The order is standard, 
and the angles are positive when the bomber makes 


c In S " = SS' the product matrix S" is computed by the 
row by column rule: to find the element in the zth row and jth 
column of S ", multiply the elements in the ith row of S by 
the corresponding elements in the jth column of S' and add. 
Note how this rule is applied in the text to multiply a 3 X 3 
matrix by a 1 X 3 matrix to obtain a 1 X 3 matrix. 


a diving banking turn to the right.) The azimuth and 
elevation of the target with respect to the air mass 
system are required. The problem is solved by the 
matrix equation 

X = SrSpSpX, 
when written in the form 


S-rX — SpSrX, 

where 



sin A cos E 


sin A cos E 

X = 

cos A cos E 

> x = 

cos A cos E 


sin E 


sin E 



cos Y sin Y 0 


~1 0 0 

Sr = 

— sin Y cos Y 0 

, Sp = 

0 cos P sin P 


0 0 1_ 


_0 — sin P cos P_ 


S R 


cos R 0 sin R 
0 1 0 
— sin R 0 cos R 


After simple calculation, one finds that 

sin E = cos P cos P[tan E cos R — sin R sin A 
— tan P cos A], 

sin (A — Y) = cos E sec P[cos R sin A + sin R 
tan E~\. 

In this problem the angles Y, P, and R are known 
from gyro readings, 97 so that these formulas are com¬ 
pletely determinate. 

In assessment programs a large number of read¬ 
ings must be reduced. An exact mathematical solu¬ 
tion may be too time-consuming. Hence, problems 
such as the preceding one are frequently solved by 
mechanisms or special computers. A mechanism 
called a glook , 34 constructed with precision and con¬ 
sisting of rotating arcs which may be positioned to 
mimic the orientation of the aircraft with respect to 
the air mass, can be used for these reductions. A 
transit is provided for exact settings and markings. 
The most appropriate computers are based on the 
principle of gnomonic projection. 54171 Using the 
center of a sphere as the center of projection and an 
equatorial tangent plane as the plane of projection, 
the great polar circles of longitude project into a 
family of straight lines and the small circles of lati¬ 
tude project into hyperbolic arcs. d The principle on 
which the use of the chart is based is simple. Imagine 
that a line from the sphere center to the target is 
fixed. Rotate the sphere and its attached planar grid 


d The U. S. Coast and Geodetic Survey has prepared plates 
for large, fine-scale (10-minute spacing) gnomonic charts. 


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REDUCTION OF DATA 


99 


at pleasure. The target line will trace a certain pat¬ 
tern on the chart. To be specific let the grid plane 
have its normal line at an elevation of 0° and an 
azimuth of 270°, then, if a yaw is to be removed, the 
sphere rotates in azimuth and the trace moves along 
the proper hyperbola (line of equal elevation); if a 
pitch is to be removed, the sphere rotates about the 
normal line of the grid and the trace actually de¬ 
scribes a circular arc on the grid centered at its 
center; and if a roll is to be removed, the trace moves 
along a small circle on the sphere which projects into 
one hyperbolic arc of a family orthogonal to the 
original set. Instruments based on these and similar 
projection principles have been constructed. 27 28 

7.3.3 Calculation of Deflection 
and Parallax 

The calculation of the deflection that should have 
been taken is carried out by direct application of 
elementary ideas. In particular, kinematic deflection 
is determined by the classic timeback method. If it is 
assumed that the gun mount is in nonaccelerated 
motion, the range at any instant can be called a 
future range, and it may be supposed that a bullet 
fired earlier under perfect aim is just hitting the 
target. The dependence of time of flight on the orien¬ 
tation of the gun barrel is insensitive enough to 
bearing changes so that the azimuth and elevation of 
the target at the instant of supposed impact may be 
used. (These differ from the true bore angles by 
ballistic deflections only.) Entering prepared tables 
or using appropriate graphs for the known airspeed 
and ballistic density yields the time of flight. Pro¬ 
ceeding back along the motion picture record by this 
length of time gives the target position at the instant 
of fire. The differences in stabilized azimuth and ele¬ 
vation of the two target positions give the com¬ 
ponents of kinematic deflection. (We need not labor 
the evident details of ballistic corrections, except 
to point out the necessity of special charts 34 and 
computing aids in such mass production operations.) 
The correct deflections may then be plotted against 
odd times (assessment times minus time of flight) and 
a curve may be smoothed through these points. 
The advantage of this elementary timeback idea is 
that full allowance is made for target acceleration. 
Allowance for the acceleration of the center of gravity 
of the gun mount is much more difficult. 115 - 172 To 
date, such allowance has been considered unnecessary 


since violent avoiding action has not been seriously 
studied. 

A parallax correction in the bearing of the target 
must usually be made because of the appreciable dis¬ 
placement of gun position from that of the recording 
instrument. In fact, if a, b, c are the coordinates of 
the gun relative to the camera in the aircraft coordi¬ 
nate system, then the azimuth and elevation of the 
target relative to the turret may be called A + A A 
and E + A E, and, when the range is r from the turret 
and r 0 from the camera, 

r sin (A + AA) cos ( E + A E) = r 0 sin A cos E — a 

r cos ( A + AA) cos (E + A E) = r 0 cos A cos E — b 

r sin {E + A E) = r 0 sin E — c. 

From these equations the approximations 

. b sin A — a cos A 

A A =-—- » 

r cos E 

{a sin A + b cos A) sin E — c cos E 

A E =- • 

r 


are obtained. In practical work, a and c may normally 
be neglected and suitable charts may be prepared 
from which the correction may be read. As an 
alternate to the formulas, the parallax correction may 
be made by another mechanism with an outre name 
— the plaxie . 34 

7.3.4 Effect of Errors in Measure¬ 
ment and Single-Shot 
Hit Probabilities 

Indirectly, the discussion above indicates that re¬ 
duction of the raw data has been brought to the point 
where errors in gun aim are known as functions of 
time over an attack. The next question is a natural 
one: if a single bullet is fired at a given instant with 
such false aim what is its probability of hitting the 
target? And, upon deeper consideration, how is this 
probability affected by the measurement error com¬ 
mitted in stating the gun error? (Such errors in meas¬ 
urement must arise because of uncertainties through¬ 
out the process in observation, machine behavior, 
recording, and calculation.) 

The first question may be answered initially on the 
assumption that there is no measurement error. 98 - 116 
Certain additional assumptions about the target and 
about bullet dispersion are made before proceeding 
with the solution. The target is unconditionally vul- 


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ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


nerable. For such a target the probability that a hit is 
lethal depends on the position of the hit but is inde¬ 
pendent of the number or location of earlier hits. 
Abundant evidence, both theoretical and empirical 
in character, indicates that this is a valid assumption 
to make about fighter aircraft. It is recommended 
further, 98 that the typical fighter target be replaced 
by a sphere of diameter 5 ft with a uniform vulner¬ 
ability factor of 0.40. This means that if a bullet hits 
the previously undestroyed sphere at any point the 
probability that it will be lethal is 0.40. Turning to 
the bullet, angular displacements may be measured 
in a plane of action (the plane containing target and 
gun-mount velocity at a given instant) and perpen¬ 
dicular to this plane. Thus, if a perfect bullet with 
no dispersion were fired with a given aiming error, the 
relative angular separation from target center to the 
bullet when both are at equal range may be called 
(z) = ( x h x 2 ). The gun error generated by the test is 
(:r). An actual bullet fired with gun error ( x ) will 
have a relative angular separation from the target 
center, when both are at the same range, of 
(x + 6) = (Xi + bi, x 2 + 6 2 ), where (6) is a measure 
of dispersion. (Strictly, the actual bullet is assumed 
to have the muzzle velocity and initial yaw of the 
perfect bullet for reasons of range matching.) It is 
supposed that (6) is distributed normally with a 
mean of (0,0) and a variance of (of) = (of,, of 2 ). The 
quantity (6) includes the effect of firing at a fixed 
target as well as the excess in aim disturbance in 
combat over the disturbance in a test. The variance 
(of) is called the gun-mount variance. 

With these assumptions it is evident that the 
actual bullet will hit the spherical target if 

(xi + 61) 2 + ( X2 + 62) 2 < p 2 , 

where p is the angular radius of the target at the in¬ 
stant when the bullet is at the same range. In other 
words, the probability that the bullet will hit the 
target is the volume, under the two-dimensional 
normal dispersion surface, erected on a circular base 
in the (61,62) plane of radius p and center (—Xi, — x 2 ). 
The single-shot lethal probability is, consequently, 

p = 0.4 f f Q —- 

J Jt ^71-0-6,0-62 

0AG(Xi, X 2 , p, 0"6,, 0"6 2 ), 
where T is defined by 

u\ + u\ < p 2 . 

The first question, as to the single-shot probability, 
is answered completely if this calculation is refined 


to permit variations in muzzle velocity and initial 
yaw. 121 The refinement is accomplished by setting 
6 = c + 6 1 , 

where c is quasi-steady and 6 1 is fluctuating (in the 
sense of the second classification of errors of Sec¬ 
tion 7.1.2). 

The effect 98 of a measurement variance (of,) may 
now be introduced. Instead of knowing (x), one 
knows only (z), where 

zi = X! + Vi 
Z 2 = X 2 + 2 / 2 . 

The total measurement error (y) may, in its turn, be 
written as a sum of a quasi-steady part e ( h ) and a 
fluctuating part (m). It is the fluctuating part which 
has the variance (o-«). The problem is to estimate 
the lethal probability p as adequately as possible 
using (z) and (a 2 ). The estimate of p selected is 
p' = 0.4(r(zi, z 2 ; p; 0-1, o- 2 ), 

where 

2 2 2 
O" 1 0"6, 0" m , 

02 = of 2 - OW 

(The subtraction of variances should be carefully 
noted.) The sense in which p' is an adequate estimate 
of p is that its weighted average over the universe of 
possible ( m ) (or its expected value with respect to 
(m)), 

0 4 r +o ° r +m . 

2 J — oo «/ —oo 

G(xi + 61 + mi, x 2 + 62 + m 2 , p; ai,d 2 )dmidm 2 , 
is precisely equal to 

0 AG(xi + hi, x 2 + h 2 ; p; o-6„ o-6 2 ). 

The result, and with it, the subtraction of vari¬ 
ances come about as follows. Starting with the idea 
that the weighted average of G(z h z 2 ; p; 0-1, 0-2) should 
be G(x 1, x 2 ; p; <r bl , (r&J, when h is put equal to zero, 
the identity 

> 4 -oo /»+oo 1 2 2 1 

1 / 1 _2/0- «v. 2 /O _ i- 


O 


ml /2<Tmi — m* !2<Tm? 


1 (T m2 2 t( 7 i(T 2 


2tt V<4, + of V<7m 2 + of 

g_(ui— xi) 2 /2(<rml+<rl)— (m-xt)* /2(vml-\-o%) 

clearly requires of = o-^ + a 2 , which yields the o- 2 
described above. 


e It will not be inferred that the quasi-steady part of the 
measurement error has any relation to the quasi-steady gun¬ 
pointing error. The same remark applies to the fluctuating 
part. 


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MEASURES OF EFFECTIVENESS 


101 



000 llJl Ml 1 MM i /U_/— L—L - L - L - ^_J - ^ _ 

0 2 5 10 15 20 25 30 

6/x 

Figure 1 . Single-shot hit probabilities. 


In a poor experiment it is conceivable that 
Cm > a\. It can be shown analytically 99 that the 

above solution exists uniquely when and only when 

2 / 2 
C TO fffr . 

In practical work it is not usually necessary to 
evaluate the double integral defining p'. The assump¬ 
tion commonly made is that <ri = a 2 = a. Then, if 
the distance £ = z\ + z\ is used, the probability p' 
can be expressed as a function of p/a and £/<r. Based 
on tables made for scatter bombing, 55 charts have 
been prepared 22 which permit one to read p' (or 


rather p'/0.4) directly. Two of these charts appear 
here as Figures 1 and 2. 

7.4 MEASURES OF EFFECTIVENESS 

7.4.1 Recommended Measure and 
Others 

If an objective of an airborne assessment program 
is to rank two or more fire control systems, some 
method of compressing single-shot probabilities, 


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102 


ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 



which are given as functions of time over particular 
attack courses, should be devised. One such com¬ 
pression is the generation of the average probability 
of at least one lethal hit for a properly stratified set of 
particular attacks of the same general type. Proper 
stratification means that the replications faithfully 
permit the variables of gunner, installation, attack 


path variations, and the like to express themselves. 
This criterion of effectiveness 98 may better be ex¬ 
pressed as the expected proportion of engagements of a 
given type which will he successful for a system. Suc¬ 
cess is the destruction of the target, the survival of 
the gun mount, or some combination of both. 

Other measures of effectiveness have been pro- 


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OPTIMUM DISPERSION 


103 


posed. One might determine the expected number of 
hits on the target per engagement. Two objections 
may be raised. First, since the target is uncondition¬ 
ally vulnerable, there is no cumulative effect of non- 
lethal hits. Second, because of serial correlation in 
the gun-pointing errors during the attack, the num¬ 
ber of lethal hits cannot be converted into a destruc¬ 
tion probability. 

A common British measure has been the index I 
defined by the expression 

I ioo (Bullet ^ ens ^ y on us i n £ the system) 

(Standard bullet density) 

The standard bullet density is the density at the 
target range of the same number of bullets (as in the 
numerator) distributed uniformly over a circle 1° in 
diameter. With the aid of /, determined by experi¬ 
mental procedures, hit expectations and the proba¬ 
bility of success in an engagement have been calcu¬ 
lated. 199 This measure, and others 118 have been care¬ 
fully reviewed. 126 

Although the discussion here gives insufficient de¬ 
tail to make it evident, the problem of measures of 
effectiveness is sophisticated. One can be led to 
speculations about the relative worth of systems 
whose performances as a function of long- and short- 
range invert. One can attempt to predict expected 
losses and gains in combat, e.g., fighters shot down 
per bomber lost, similar to those determined after 
the event from an analysis of combat records. 165 It 
is likely that these subtleties are inappropriate in the 
current state of the art. But the attempt to relate a 
a particular technical branch of air warfare to the 
overall situation may well be prosecuted in the 
future. 

7.4.2 Expected Proportion of 
Successful Engagements 

Turning now to the first criterion, the expected 
proportion of engagements which will be successful, 
suppose that shots are fired at times ti, t 2 , h * • • dur¬ 
ing a particular run. The probability that the target 
will survive the first i shots is 

m = [i - v \h) ][i - P '(«] •••[!- v'm. 
Consequently, the probability that the target will be 
destroyed during the first i shots is 

d(ti) = 1 — s(ti ), 

provided that the bomber is invulnerable in the sense 
that it can fire these i shots. Since we wish to work 


with a type of tactical situation rather than a par¬ 
ticular attack of that type, the expected value, D(ti), 
of d(ti) is required. One may think of D(U) as re¬ 
ferring to an engagement involving a particular type 
of attack and type of fire control, but with a random 
selection of fighter pilots, gunners, equipment speci¬ 
mens, weather, and like variables. 

There are two ways of estimating D(t). The direct 
method determines d(t) for each attack of the type 
and forms an average of the resulting values. The 
statistical method attempts to describe the universe 
of gun errors, made during all attacks of the type, by 
certain parameters. D(t) is to be estimated from these 
parameters. For example, it may be assumed that 
gun errors can be described as a two-dimensional 
Gaussian distribution with the parameters: mean 
traverse error, mean elevation error, variance of 
traverse error, variance of elevation error, correla¬ 
tion coefficient of traverse and elevation errors. The 
adequacy of this particular description may be 
questioned on the ground that the effect of serial cor¬ 
relation in aim wander during a representative attack 
has been ignored. If, however, the universe of gun 
errors can be correctly described, then the statistical 
method is superior to the direct method in that it is 
statistically efficient, extracting the maximum amount 
of information from the available data. 

The value of D(t) as a measure of effectiveness does 
not depend on the assumption of bomber invulner¬ 
ability. If A and B are two fire control systems, for 
which Da(£) and D B (t) are known, and if Zhi(£) is 
greater than D B {t) for all values of t, then: A is 
superior to B in the sense that A will enjoy a greater 
proportion of successes than B whether success is 
defined as the destruction of a target by an actual 
(vulnerable) bomber or as the survival of an actual 
(vulnerable) bomber. The theorem requires detailed 
proof 98 which will not be reproduced. 

7.5 OPTIMUM DISPERSION 

7.5.1 Expectation of Optimum 

Pattern Size 

In aerial gunnery, dispersion is essential if hits are 
to be scored. The reason is that, with these weapons, 
a burst is fired with a certain bias in aim. The mean 
point of impact [MPI] is displaced from the center 
of the small target and, when the MPI is off target, 
zero dispersion means that no hits are obtained. 
With the aiming errors of current systems the MPI is 


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104 


ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


rarely on target for more than a fraction of a second. 
In fact, for systems in which the MPI can be kept on 
target for an appreciable length of time, it is desirable 
to renounce the principle of rapid fire of small caliber 
projectiles in favor of slower fire of larger caliber ones. 
The vulnerable subarea of the target increases with 
caliber up to the point of complete structural failure 
caused by a hit on any part of the target. 

In the opposite direction, excessive dispersion is 
undesirable. Even with a perfectly aimed gun the 
spacing could be so large between bullets of the burst 
that the small target may escape. Large dispersions 
are used in warfare only against large targets. 

Evidently, one looks for an optimum size of the 
dispersion pattern. The determination of this size de¬ 
mands a knowledge obtained from experimental pro¬ 
grams of the universe of gun-pointing errors, al¬ 
though it is possible to attempt to use indirect combat 
information such as rounds fired per enemy aircraft 
destroyed in an analysis. 168 Such a procedure is of 
doubtful efficiency since, not knowing how the gun¬ 
pointing errors vary and how they are serially cor¬ 
related with range, it must assume a fixed target 
size and a constant aim error (or one that varies ac¬ 
cording to some supposed distribution). To optimize 
the dispersion with assurance one needs precise gun¬ 
pointing information and should use the technique 
of Section 7.4.2 to determine the average destruction 
probability. 

7 . 5.2 Nose Attacks on B-29 

The determination of optimum dispersion size ac¬ 
cording to this schedule appears attractive in con¬ 
nection with nose attacks on the B-29, 50 since ex¬ 
tremely accurate data are available from tests on the 
testing engine of the War Research Laboratory of the 
University of Texas. 1 The attacks were both pure and 
aerodynamic pursuit courses. Four modifications of 
the nose-sighting stations with twelve replications of 
each of four attacks furnished 192 samples of the 
type of tactical situation in question. Six gunners 
were used. 

The numerical values of the variances (see Sec- 


f In this machine an actual gunner operates the sighting 
mechanism against a programmed target whose motion is 
very realistic. The machine records continuously the gun 
errors in traverse and elevation. It, of course, has preknowl¬ 
edge of the correct values. As a major advantage, replications 
are possible in which only gunner and sight specimen are 
variables. Even platform motion is programmed. 


tion 7.4.2) are: g\ = 2.58 (after tests at the AAF 
Central School for Flexible Gunnery at Laredo, 
Texas); a mi = 1.16, <r^ 2 = 0.41 (estimates of machine 
accuracy at the University of Texas.) We shall take 
a = \/(ti(T2 = VVf — <7^i a/o| — (j^ 2 = 1.33 milli- 
radians. The results of a calculation of D (the average 
of all destruction probabilities with t set equal to the 
duration of the attack) as a function of < 75 , by the 
methods of Section 7.4.2, is shown in Figure 3. The 



1.07 1.61 2 3 4 5 6 7 

^ IN MILS 

Figure 3. Average probability of at least one lethal 
hit. 

figure also gives a 0.95 fiducial zone. This means that 
the probability is 0.95 that D lies in the interval 
shown for any given <76. 

It is concluded from the figure that the optimum 
dispersion is ab = 3 milliradians. However, the actual 
value <Jb = 1.61 corresponds to a destruction prob¬ 
ability of 0.25 which is only 0.03 below the maximum 
probability. About all that can be said definitely is 
that <76 should not be reduced below its present 
value. 


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OPTIMUM DISPERSION 


105 


7.5.3 Theoretical Solution for 
Fighters 

In spite of the emphasis of this chapter on the 
necessity of precise experimental results, there are 
certain advantages to an entirely theoretical ap¬ 
proach to these problems. The average assessment 
program is complicated and costly in materiel and 
personnel. Its conclusions may be reached too late 
and even then may be subject to censure on grounds 
of experimental design or accuracy. In the theoretical 
treatment the variables are always under control. 
The relative effects of variation in different param¬ 
eters may be readily studied. Such a study is also 
modest enough to state not what will happen but 
what the range of results will be for some range of 
variability in a parameter whose values cannot be 
known exactly. When its results point in a direction 
independent of such ranges of variability, it is con¬ 
clusive and an experimental study is obviated. 
Finally, the theoretical approach may lead to 
economy and efficiency in experimental programs by 
dictating exactly what details need an empirical 
foundation. 

Some of these advantages are illustrated by a study 
of the performance of a fighter aircraft (an F6F) 
directed by a gyro gunsight (the Mark 23) , 38 The 
fighter attacks a target flying straight and level at 
constant speed. To determine the basic position and 
orientation of the fighter at each instant, it is as¬ 
sumed first that a perfect pursuit curve' is flown. To 
get single-shot probabilities this orientation is per¬ 
turbed by the ranging error assumptions (1) perfect 
range, (2) consistent under range by 10 per cent, 
(3) consistent over range by 10 per cent, and (4) 
varying from 25 per cent under range at 800 yd to 
25 per cent over range at 200 yd; and by the tracking 
assumptions of (1) perfect tracking, and (2) tracking 
with a circular normal distribution of errors so that 
the pipper is within 3 milliradians of the target 
during 50 per cent of the time. (Such ranging and 
tracking assumptions may come from rapid pre¬ 
liminary experimentation or even from subjective 
estimates by operators.) In addition to these oper¬ 
ational errors, Class B errors will occur because of 
the particular calibration of time of flight in the 
sight. In fact, if A* is the correct lead, the sight’s 
time-of-flight multiplier should be 


where 0 is the approach angle measured at the 


bomber. (This is in agreement with the calibration 
ideas of Section 5.3.5.) At a given altitude, tem¬ 
perature, range, and angular rate, the sight will 
actually use a value t m according to the calibration 
of the range circuit. Then the error in lead is ap¬ 
proximated by 

* v B sin (6 — A*) 

€ A *m) ) 

r 

which is the error in time of flight multiplied by the 
angular rate of the gun-target line for a correctly 
flown fighter. This expression for the error includes 
the effect of lag in lead computation since t m was 
chosen with attention to the equation of the sight. 
The remaining sources of Class B error, such as 
optical and electromagnetic dips, and feedbacks, are 
ignored. The performance of the fighter really ap¬ 
pears through t m in the above expression. 

The method of computing single-shot probabilities 
is relatively straightforward. The value of e A gives 
the displacement of the mean point of impact from 
the target. This mean point of impact is surrounded 
by a circular normal distribution whose variance is 
the sum of those due to dispersion and aim wander. 
To get this sum it is assumed that a good value of 
<7b (arising as a result of ground firings of a fighter- 
mounted caliber .50 machine gun using API MS 
ammunition) is 1.2 milliradians. Next, the gun point¬ 
ing is an amplified form of the sight line wander (see 
Section 5.3.4, u ‘Amplification of Tracking Errors”) 
which depends on the frequency. An average value 
of 1.5 is selected arbitrarily for the amplification 
factor. Consequently, the variance due to gun 
wander is [(19) (0.425)] 2 , where 0.425 converts the 
diameter of a circle containing 50 per cent of the 
random gun directions into a for this distribution 
of directions. Finally, the pattern is narrowed in the 
air because of the forward motion of the fighter (see 
Section 1.5.3) by a factor v q /{vq + vf)- If v F = 209 
yd per sec and v 0 = 956.7 yd per sec, the shrinking 
factor is 0.82. The shrunk sum of variances is 10.85 
milliradians squared. The target is taken to be a 
small bomber represented by a circle of radius 1 yd 
with a uniform vulnerability factor of 0.3. The 
single-shot probabilities may now be read from 
Figures 1 and 2. 

In fighter fire a complication arises because of the 
harmonization of a battery of guns. A typical U. S. 
Navy procedure is to boresight the six wing guns to 
converge horizontally and vertically with the line of 
sight at a range of 300 yd. Strictly, one should there- 


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106 


ANALYTICAL ASPECTS OF AIRBORNE EXPERIMENTAL PROGRAMS 


fore calculate probabilities for each gun separately 
or at least for the middle port gun and the middle 
starboard gun. When this is done, paying proper 
attention to parallax, boresighting, and aircraft 
banking, it is found that the probabilities for the 
various guns do differ appreciably. In summarizing 
calculations the joint effect of all guns firing together 
and continuously is given by totalizing the effects of 
the individual guns. 

The single-shot probabilities may be processed into 
the probability of at least one lethal hit (as a function 
of time along the course) by the methods of Section 
7.4.2. For an attack, made at 371 knots against a 
target at 195 knots, starting at 900 yd on the beam 
and terminating 3.3 sec later at a range of 200 yd and 
an angle off of about 40°, the results of the calculation 
are given in Table 1. 


Section 7.3 considers the reduction of the raw data. 
Techniques rather than schedules are given. A re¬ 
curring problem of rotation of coordinate systems is 
handled generically either by matrix methods, or by 
mimicking mechanism, or by gnomonic computer. 
The calculation of deflection uses the timeback 
method in which a given frame represents impact by 
a bullet fired earlier. Formulas for the parallax cor¬ 
rection required by the displacement of guns and 
camera are given. The final technique is considered 
in detail. This is the problem of reducing gun point¬ 
ing errors to the probability that a single round 
fired with such an error would be lethal to a fighter 
target in the sense that the fighter could not return 
to its base. It is also emphasized at this point that 
measurement errors of the experiment lead to a 
measurement variance which must be subtracted 


Table 1 . Probability of at least one lethal hit as a function of time X 100. 


Time 

(seconds) 

No range error 


Range 10 per cent 
too small 

Range 10 per cent 
too large 

Range from —25 per 
cent to +25 per cent 

<7 = 0.98 <7 

= 3.29 (7 

= 5 

<t = 0.98 

(7 = 3.29 

( 7=5 

a = 0.98 

<7 = 3.29 

< 7=5 

<7 = 0.98 

<7 = 3.29 <7 = 5 

0.6 

56 

33 

24 

0 

0 

1 

0 

3 

3 

0 

0 0 

1.2 

74 

64 

50 

0 

0 

1 

0 

3 

6 

0 

0 0 

1.8 

74 

73 

66 

0 

0 

1 

0 

24 

31 

0 

1 6 

2.4 

74 

77 

76 

0 

0 

1 

45 

85 

75 

71 

67 58 

3.3 

74 

78 

82 

0 

0 

1 

100 

100 

98 

100 

100 99 


7.6 SUMMARY 

The introduction, Section 7.1, points out that the 
basic purpose of airborne assessment programs is the 
ranking of different fire control systems. To arrive at 
changes in design, a combination of analytical 
methods and controlled tests on laboratory ma¬ 
chines is necessary. The errors of a fire control sys¬ 
tem are first classified by cause: mechanism, Class 
B, operational, and dispersive; and then by statistical 
nature: fluctuating, quasi-steady, and steady. Cur¬ 
rent assessment programs use simulated attack situ¬ 
ations with camera recording to get at the overall 
errors of systems. 

Section 7.2 discusses the methods by which the 
path in space of a target may be determined. In¬ 
strumentations are: three fixed cameras, plus a range 
camera, plus gyro records of platform motions; a de- 
flectometer which records the turret position in 
azimuth and elevation; long focal length cameras 
fixed on the ground with data reduced by astrometric 
methods; a camera in the gun mount and also one in 
the target; and, simply, one camera which relies on 
fixed and distant reference points in the landscape. 


from the dispersion variance before the single-shot 
probability is computed. 

Section 7.4 is concerned with a still further reduc¬ 
tion, the generation of a measure of effectiveness 
from the single-shot probabilities. The measure 
proposed is: the expected proportion of engage¬ 
ments of a given type which will be successful in the 
sense that the target is shot down, or the bomber 
survives, or some combination of these occurs. This 
measure is the evolutionary result of much previous 
thought. 

Section 7.5 discusses the choice of an optimum size 
for the dispersion pattern in relation to aiming 
errors. Using experimental data of a precise nature, 
the situation for nose attacks on a B-29 is explored 
by the schedule of the previous section. Using purely 
theoretical methods, the situation for a fighter 
equipped with a gyro gunsight is discussed. For the 
first case, pattern size might be increased slightly 
over its present value, but the change in probability 
of at least one lethal hit would be only from 0.25 to 
0.28. For the second case, it is shown that errors in 
ranging greatly overshadow variations in dispersion. 


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( 

Chapter 8 

NEW DEVELOPMENTS 


8.1 INTRODUCTION 

T his chapter does not predict future develop¬ 
ments. Instead it describes in compact form 
those systems and ideas which were being seriously 
considered in the closing months of World War II 
as suitable for the next stage of that war. As a result, 
the ideas are, on the whole, limited in perspective to 
the air tactics then current. The fire control systems 
discussed are concerned with doing a better job than 
existing equipment. But it is the same job. For ex¬ 
ample, no real step is taken to solve the problem of 
an ultra-speed target, nor are ranges of combat ex¬ 
tended materially. Under new tactics, only rather 
slight variations in existing fighters are considered. 

These remarks are critical rather than censorious. 
Failing the introduction of unforeseen and radical 
principles, the systems of the future will probably lean 
heavily on the ideas of this chapter. Consequently, 
these ideas must be explored still further in order to 
gain an adequate knowledge of theoretical and me¬ 
chanical limitations. The best case in point is radar 
gun laying, which involves both the theory of track¬ 
ing and the facts of mechanical responses. 

One may hope that the next phase in fire control 
development will consist of a fresh reexamination of 
the problem, which will result in a system combining 
the best of the new ideas and which will avoid the 
known errors of existing mechanizations. The theo¬ 
retical design of such a system should precede its 
construction, for it is a curious fact that many 
machines are built and then analyzed, with patch- 
work as the usual result. 

Fire control design has lagged tactical circum¬ 
stances. If the broad outlines of future combat cir¬ 
cumstances are made available to this design, even 
at the early and tentative stages of tactical expecta¬ 
tion, this need not be true. 

8.2 STABILIZED SIGHT SYSTEMS 

8.2.1 Nature and Types of 

Stabilization 

Of the various mutations that have occurred in the 
development of airborne fire control, stabilization is 


perhaps as important as was the introduction of 
power turrets. Stabilization permits the gunner to 
concentrate on his real job, tracking the smooth 
target motion. The normal irregular motions of the 
gun platform, which make this a difficult task under 
the usual arrangement, no longer disturb the gunner’s 
line of sight. In principle, any motions not called for 
by the gunner through his handlebars are detected by 
gyroscopes which immediately send impulses to the 
turret motor which, in turn, acts to neutralize the 
perturbation. 

Either position or rate stabilization may be used. 
In position stabilization a free gyroscope is employed. 
By the first basic principle of gyroscopic behavior, 
i.e., maintenance of a fixed direction in space, motion 
of the sight line away from the gyro axis is detected 
and corrected when the handlebars are in neutral. In 
tracking, a connection between handlebars and gyro 
is made so that the gyro will precess when moved by 
the handlebars. It is the differential signal, between 
handlebar motion and total motion, which is de¬ 
tected and eliminated by the stabilization. In rate 
stabilization, the gyro is constrained. When the line 
of sight moves, taking the gyro with it, the gyro re¬ 
sponds with a precessional torque (kick). The mag¬ 
nitude of this torque is proportional to the angular 
rate of turn of the gyro. If the sight line starts to 
move, under external influences, the precessional 
torque of the gyro is measured and the turret motors 
restore the sight line to its original position very 
quickly. When tracking, the gyros must be prevented 
from stabilizing out the intended motion. Since the 
gyro is constrained, no major precession occurs. The 
kick due to the handlebar motion is neutralized by 
suitably calculated impulses from the handlebars. 
Thus, this part of the kick does not act to restore the 
turret to its original position. 

It will be appreciated that there is nothing inherent 
in stabilization which will improve the mechanism 
on which lead computation is based. Instead, one 
expects the improvement in inputs to cause the com¬ 
puter to perform better. The stabilization circuit 
may be regarded as an improved analogue of the 
B-29 follow-up system under which the guns follow 


CONFIDENTIAL 


107 


108 


NEW DEVELOPMENTS 



Figure 1 . Unit schematic, Type S-4 sight. 


the sight line. The lead computation is an additional 
function in each case. 

8.2.2 Fairchild S -4 System 

An excellent example of a rate stabilized sighting 
system is the Fairchild S-4 (or S-3) installation 220 for 
inhabited turrets. The stabilization features are the 
same for both types. The S-4 is ballistically superior 
in that it will function for all types of Type 5 pro¬ 
jectiles in a speed range for which the three-halves 
power drag law is valid. One need only insert muzzle 
velocity and ballistic coefficient. A ballistic mecha¬ 
nism then effectively solves the Siacci differential 
equations for the trajectory, instead of translating 
special ballistic tables calculated by this method and 
referring to particular ammunition. (This is an excel¬ 
lent example of how proper fire control design can 
anticipate ordnance changes and thus avoid lengthy 
and expensive modification in the future.) The S-4 
will be discussed. The theory developed will also 
cover both hydraulic and electric turret mechanisms. 

There are two major circuits of the system. One 
operates the azimuth turret motor and controls the 
lateral lead. The other operates the guns in elevation 
and controls the vertical lead. Each of these circuits 
employs a captured gyroscope which has a dual role. 
It enters into the determination of the kinematic lead 
of its component and, independently of this, stabi¬ 
lizes the line of sight in its component. The unit 
schematic of the S-4 is shown in Figure 1. 

8.2.3 Circuit Components of the S -4 

The system may be best understood by writing 
out the differential equations of the circuit. 100 It 


suffices here to consider the situation in one com¬ 
ponent. The general picture involves the interaction 
of two components and is necessarily complicated. 
The basic assumption of this treatment is that the 
inertia of the various moving elements may be 
neglected. This is a major assumption since success¬ 
ful operation depends on the relative speed of re¬ 
sponse of gyros, servomotors, 3 and the amplidyne 
turret. 

Let a gyroscope whose spin axis lies in a horizontal 
plane be rotated about an axis perpendicular to that 
plane, i.e. let it track in azimuth only. Then the 
gyroscope develops an internal torque whose axis is 
perpendicular to the spin axis and to the axis of rota¬ 
tion and so also lies in the horizontal plane. The 
magnitude of this torque is I^lda/dt, where / is the 
moment of inertia of the rotor, 12 is the angular speed 
of the rotor, and a is the angle at any instant between 
the line of sight (the spin axis) and some direction 
fixed in space, i.e. the direction of a star. If the gyro 
is so mounted that the spin axis can move only in a 
plane vertical to the horizontal it will precess in this 
vertical plane by an angle 6. If its motion in the 
vertical plane is inhibited by a damping torque 
proportional to velocity, Ddd/dt, and a spring torque 
proportional to displacement, SO, then 


since all three torques have the same instantaneous 
axis in the horizontal plane. 

a A servomechanism is a power control device in which a 
weak input is amplified to control a strong output. This out¬ 
put, in turn, feeds back to combine with and modify the 
original input. 


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STABILIZED SIGHT SYSTEMS 


109 


The displacement angle 9 can be transformed into 
an output voltage K\9 by means of an E magnet. 
This device is illustrated in Figure 2. When an a-c 
voltage is impressed on the primary the output from 
the secondary is zero, when the gyro is undeflected, 
since the output arms are wound in series opposition. 
Upon deflection, more voltage is induced in one arm 
than in the other, and the arrangement is such that 
the imbalance is proportional to the deflection. 



Figure 2. Gyro E magnet unit. (Courtesy of Fair- 
child Instrument Co.) 


The voltage from the hand control potentiometer 
is K 2 p, where p is the positional angle of that po¬ 
tentiometer. Because of the interconnection (8) (see 
Figure 1), this voltage is subtracted from the total 
voltage (2) Kid. Hence the input to the stabilization 
servo is ei = K x 9 — K 2 p. The output voltage (3) e 2 of 
the servo is tapped off by a moving potentiometer 
arm so that 


The inputs to the electrical differential (two poten¬ 
tiometers in parallel) are (3) e 2 , and (9) Kap from the 
hand control potentiometer. Hence the output (4) is 
a voltage e 3 such that 


to the original reference line in space. (The angle a is 
the unwanted angle of yaw.) Consequently, 


d(y — a) 
dt 


= K,e 


Up to this point nothing has been said about the 
total lead angle A. Since the sight head is affixed to 
the turret, the sight line will follow roughly the gun 
line. The small displacement of sight line from gun 
line to give deflection is accomplished by a sight 
servomotor whose mechanical output gives a rate to 
the sight line relative to the sight head which is 
proportional to its input voltage. Now the computer 
produces two voltages representative of the time-of- 
flight multiplier t m and the ballistic deflection A&. 
And in a steady state condition the rate of rotation 
of the sight line is proportional to the control handle 
potentiometer position, or 


da 

dt = ^ 


Hence introduce as the sight servomotor voltage in¬ 
put Kit m p — A b — A. The output of the sight-line 
motor is a rate of change of sight line relative to the 
sight head (i.e., relative to the gun line), and this 
rate d(y — <j)/dt = dA/dt is proportional to the volt¬ 
age input. Hence 
T _ dA 

K& — = Kit mil — Ab — A . 
at 

It follows immediately that there is a delay in lead 
computation for this system. (See the analysis of 
Section 5.3.4 “Delay in Lead Computation.”) Only 
if A is constant and equal to Kit m p — A b will the servo¬ 
motor stop. 

This account rather conceals the role of the gyro 
in measuring angular rate. Actually the whole system 
acts to keep the input voltage e\ to the stabilization 
servo zero, or to make K x 9 = K 2 p, this being the 
essence of stabilization. In other words the voltage p 
being used in determining lead is really measured by 
the precessional torque of the gyro. 


e 3 = Km — e 2 . 

Assuming that the output voltage (5) of the ampli- 
dyne is proportional to the input voltage, the angular 
rate given to the guns by the turret motor is propor¬ 
tional to this amplidyne output voltage. But the 
angle which the guns make with the longitudinal axis 
of the aircraft is y — a where y and a are the posi¬ 
tional angles of gun and longitudinal axis with respect 


3.2.4 Complete Circuit Equations 
of the S-4 

The preceding basic circuit equations are combined 
most expeditiously by placing d/dt = p and treating 
p as an algebraic quantity. There result 

TQK k k 

P 2 (y - a) = _ 1 3 l pa + K b (K 2 K 3 + Ktp)p (1) 

Dp + S 


CONFIDENTIAL 






































110 


NEW DEVELOPMENTS 


' o , IOKiK&p} 

?+ D^rr) y = pa+ ~5^s 

K 7 t m H — A b 


V 2 + 


1 + K&P 

mK x K,K, V \ . K 7 

)cr = p 2 a-p 2 — 


MK^sKsp 

Dp + S 
+ K b (K 2 Ks + K*p)p (2) 


K 7 t m fj. A b 


Dp + S 


+ Kqp 

+ K h (K*K* + K,p)p. (3) 


Since the yaw a of the aircraft enters the equation as 
d 2 a/dt 2 , it follows hat the S-4 stabilizes completely 
only nonaccelerated motions. Again, a rather bizarre 
form of aided tracking is present since by equation 
(2) neither gun position nor gun rate is simply pro¬ 
portional to handlebar position. 

To complete this phase of the discussion a prime 
advantage of this system should be emphasized. The 
notion of time constant or decay of false leads does 
not apply except in the generalized sense of the time 
for unwanted transients to vanish. The point has not 
been made out in detail theoretically. But from the 
standpoint of actual operation, the gunner seems to 
have about as much control over his line of sight as 
he would have with a fixed sight. The disturbed 
reticle behavior is not in evidence. 


8.2.5 Class B Errors of the S-4 

Little can and will be said about the Class B errors 
of the S-4. It is evident that lag in lead computation 
exists and that this lead is itself based on first order 
theory 101 and does not allow for curvature. Calibra¬ 
tion is necessary. There is also a small error com¬ 
mitted in computing vertical lead 102 (of the gun-roll 
type) because the vertical gyro measures an angular 
rate about an axis that is not perpendicular to the 
line of sight. This second order error is given by 
lOOOAf tan E milliradians, where Al is the lateral 
lead in radians and E is the elevation. 103 

8.2.6 Sperry S-8B System 

An example of a position stabilized system is the 
Sperry S-8B sight. For purposes of exposition the 
discussion will again be restricted to the azimuth 
plane. The appropriate sight schematic is given in 
Figure 3. Various angles needed in the development 
are shown in Figure 4. 

The angle a is the yaw of the airplane, and \i is 


interpreted as the displacement of the handlebar 
controls. The optical system is so designed that 

o- - Q = ip. (4) 

(Recalling Sections 5.3.3 and 5.5.4, we immediately 
expect to find equations resembling those for simple 
lead computing sights with a sight parameter of }/%). 
The gunner has direct control over the relative posi¬ 
tion of sight and gyro lines. The combining glass is 
controlled by the handlebars and the stabilization 
mirror by both the gyroscope and the handlebars. 



Figure 3. Unit schematic, Type S-8B sight. 



The gyroscope is universally gimbaled. It is pre- 
cessed in azimuth by a torque caused by a spring 
wound around an axis lying in the horizontal plane 
and making a 90° angle with the bore axis. This 
torque T is equivalent to a force T sec (y — 0) ap¬ 
plied perpendicularly to the azimuth plane at a 


CONFIDENTIAL 



























STABILIZED SIGHT SYSTEMS 


111 


point unit distance from 0 out along the gyro spin 
axis. Hence, as in Section 5.5.2, the precessional rate 
is 


e = 


T sec (7 — 0) 
712 


( 5 ) 


It is the function of the torque computer to make 




( 6 ) 


where X is given by 

M = X — A b 


(At being the ballistic deflection) and where t m is a 
computed time of flight. The function 4/x/3) has 
the form 


(^) = 1 +a 2 

4^ 

2 

4m 

\ 3 / 

3 


3 


where 104> 207 


a 2 = 0.0836296 
a 3 = 0.11197 


when 4/x/3 is in radians. From its form, it is evident 
that g is a correction function. Ideally, the gyro 
should precess with an angular velocity 



(See Section 5.5.2 and remember that, here, 7 — 0 = 
— <r), and 0 = 'Her.) For tracking in general, i.e., 
not simply in azimuth, this would require the torque 
T to depend on the azimuth and elevation of the gyro 
axis. To simplify the mechanism, it was decided to 
make T depend only on azimuth. Hence a correction 
function was chosen to minimize the error due to this 
design, and to distribute the remaining error as 
symmetrically as possible about the bore axis. 

Suppose now that a target is being tracked and 
that the gun mount is yawing. The output of the 
gyro is the angle of deflection 7 — 0. This output is 
first used to displace the stabilization mirror through 
this angle. At the same time the handlebar motion 
displaces the stabilization mirror by an angle 4^/3. 
(See the gear-differential circuit of Figure 3.) Conse¬ 
quently if the mirror is rotated through 3^[ 7 — 0 — 
4ju/3] and the combining glass is rotated through y/2 
(no gear reduction) it follows that the line of sight 
will be deflected by 34$ M according to the doubling 
principle of mirror rotation (Section 5.5.4). The out¬ 
put of the gyro is used secondly to displace the arma¬ 
ture of the E magnet (see Figure 2). But the gunner 
has rotated the transformer of the E magnet by 


4ju/3. Hence the angle at the E magnet is 4ju/3 — 
(7 — 0) which is y — A by Figure 4. This is the error 
signal. For small displacements, the E-magnet volt¬ 
age is proportional to this error signal. Turret servos 
are activated to move the turret in such a direction 
as to decrease this error. 

To see why we want y = A, differentiate a — 0 = 
My and use the relation 

. M + At 

v — y ’ 

tm 

in which 

* cos (7 - 0) 

'"“TO)' 

to obtain 

itmH + M = lm<r ~ Aft. (7) 

If € = y — A is the error, we have 

it m A + A = t m a — Aft — \t m e — e. 

If A is to be a smoothed value of £* <7 — A &, as it 
should be, e must be kept at zero. 

It follows that the accuracy of the lead obtained 
depends very heavily on the characteristics of the 
turret servo. Instead of subjecting the servo system 
to a componential analysis as was done in Sections 
8.2.3 and 8.2.4, one may proceed experimentally. 
This has the advantage of including inertia effects in 
the system. It has the disadvantage that the con¬ 
stants and form must be obtained as suitable aver¬ 
ages for an appreciable sample. This is not possible 
during design and construction of pilot models. To 
obtain the transit function H(p) of the system 
(p is d/dt) the definition 

A = H(p)Q, ~ A), (8) 

is used, and, typically, it is found that 

= a2p2 + ai ? + 1 , 

P 6 4 p 4 + b 3 p 3 + b 2 p 2 + bip ’ 

where a 2 = 1.161 X 10 -2 sec 2 , = 0.464 sec, 64 = 
3.84 X 10 -5 sec 4 , 63 = 2.93 X 10 -3 sec 3 , b 2 = 2.47 X 
10 -2 sec 2 , bi = 9.54 X 10 -3 sec. 

Taking H(p) = P(p)/Q(p), where P and Q are 
polynomials, manipulation yields 104 the lead equa¬ 
tion 

S(p)[P(p) + <2(P)]A = P(p)lCp° - AJ - S(p)- 

Q(p)a + S(p)Q(p)oc, (9) 

where S(p) = y&t^p + 1, and C is held constant. 


CONFIDENTIAL 













112 


NEW DEVELOPMENTS 


This is the analogue of equation (2), or equation (3), 
for this sight. 

8.2.7 Errors Caused by Aircraft 
Accelerations in the S-8B 

The presence of the term involving the yaw of the 
gun mount, a, in equation (9) is an error. How serious 
it is depends on the servo system. Typical calcula¬ 
tions show that if the gun mount is making a flat 
turn at a rate of 150 milliradians per second the 
error will be less than 1.5 milliradians. Suppose, 
however, that the aircraft is yawing harmonically 
so that 

a = A cos 27 rft. 

After transients have decayed the gun error, e a settles 
down to 

€ a = B cos (27 rft + A). 

The amplification B/A is tabulated below. 104 The 
operator H(p) was not used. The particular results 


Table 1. Amplification of aircraft motion. 


Frequency, / 

(cycles per second) 

Amplification B/A 

1st experiment 2nd experiment 

0.1 

0.012 

0.012 

0.2 

0.031 

0.036 

0.5 

0.160 

0.147 

1.0 

0.647 

0.807 

1.5 

1.396 

1.586 

2.0 

2.406 

2.363 

3.0 

2.430 

2.377 

5.0 

1.283 

1.066 


of two experiments were employed in the calculations. 
The steady-state error would be obtained otherwise 
from 

DP(p) + Q(p)> b = Q(p)cc. 

To get the transient error one solves 

ZP(p) + Q(p)Ik = 0, 

and finds that the time constant is 0.4 sec, i.e. by that 
time the transient will be less than 1/e of its initial 
value. 

The Class B errors of the system will not be pur¬ 
sued further. 207 

8.2.8 Summary of Advantages of 
the S-8B 

In summary, the main advantages of the S-8 sys¬ 
tem are (1) lead control (the handlebars control the 


lead and not the turret velocity — the disturbed 
reticle principle is inoperative), (2) variable aided 
tracking 209 (neglecting A&, from equation (7), 
a = + (1 /t m )ti, so that velocity control of the 

line of sight is (l/t m )n, and is the positional con¬ 
trol of the line of sight — recalling Section 5.3.4, 
“Aided Tracking’ 1 ), (3) improved operational sta¬ 
bility (if the handlebars are given a jerk of size /z 0 
and then held at that value, the sight line responds 
with an equally sudden jerk in the same direction of 
size fio/S and then climbs along a line of slope 
1 /t m )j and — of course — (4) full stabilization of the 
line of sight. 

8.3 RADAR 

8.3.1 Desirability and Requirements 

When the range and positional inputs to a fire 
control system are in serious error, it is only by a 
fortuitous combination of those errors that the gun 
will be pointed properly. Repeated tests show the 
real difficulty experienced by a human gunner 
when he attempts, as simultaneous operations, fram¬ 
ing, tracking, and triggering. This suggests that 
radar inputs be used to accomplish automatically one 
or two of these operations. Since framing is generally 
much poorer and more biased than tracking (it being 
difficult to frame, for example, when the target is 
not centered) and since radar ranging involves the 
simplest type of radar equipment, it seems desirable 
to use radar range input for seen fire. In the case of 
unseen fire it is essential, of course, that both range 
and rate be supplied by radar. 

Obvious requirements for airborne radar gunnery 
aids are lightness and noninterference with aircraft 
speed. Ranging equipment weighs approximately 
125 lb and is quite compact (the complete trans¬ 
mitter-receiver unit weighs 28 lb and fits in a can 
15 in. long and 9 in. in diameter). 105 The antenna ex¬ 
ternal to the aircraft is roughly equivalent to another 
machine gun barrel, and is, in fact, clamped to a 
barrel. The increased drag will cause a speed reduc¬ 
tion of only a few miles per hour. (For purposes of 
comparison, open bomb-bay doors cause a speed 
reduction of the order of 34 mph for the B-29 air¬ 
plane.) 123 

Turning to radar which tracks in angle, it might 
be presumed that an ultra-narrow beam should be 
used. The angular beam width is directly propor¬ 
tional to wavelength and inversely proportional to 


CONFIDENTIAL 







RADAR 


113 


the antenna aperture. High frequencies b would then 
be needed. Narrow beams also have the advantages 
of giving greater concentration of power (echo energy 
falls off as the fourth power of range) and of being less 
susceptible to interference (jamming). 140 However, 
ultra-narrow beams (12 milliradians) have not been 
used. Instead beam widths of 30° have been em¬ 
ployed, relying on proper antenna design for ac¬ 
curacy. For example, a dish antenna can do a conical 
scan of angle 1.5° about an axis which keeps pointing 
at the target. It is also possible to employ lobe¬ 
switching in the antenna to achieve accuracy in de¬ 
termining the direction of a target. 

The presentation of the radar output is important. 
In general, it is desirable to avoid visual presenta¬ 
tion since this requires an operator who may make 
errors in utilizing this presentation. Even with auto¬ 
matic presentation it is not desirable to consider the 
radar as purely ancillary, with a shaft rotation or 
voltage output fed into a standard computer. In¬ 
stead the original design should make provision for 
radar as an integral part of the system. Important 
simplifications may then be feasible. 159 

8.3.2 Airborne Range Only® 

An Airborne Range Only [ARO] set is the AN/ 
APG-5 {Army-Navy Airborne Pulse Gun Laying). 
APG-5 operates in the S band and has a beam width 
of 30°. Accurate pointing of the antenna at the target 
is not required. The effective tracking range is of the 
order of 1,500 yd. The range presented has a probable 
error of + 20 yd. The error is probably quasi-steady. 
That is to say, during one run it might overrange 
steadily at 10 yd, but during the next run it might 
underrange steadily at 20 yd. Such performance is 
excellent, if it can be adhered to under field con¬ 
ditions of calibration and maintenance. 

APG-5 searches automatically in range, sweeping 
from 200 to 2,000 yd twice each second. Upon en¬ 
countering a target it locks on that target and tracks 
in range automatically. The gunner must still track 
in angle and trigger. Because of ground reflection the 
present unit is sometimes not dependable under an 
altitude of 8,000 ft. 


b Nomenclature for bands is: P, 225 to 390 me; L, 390 to 
1,550 me; S, 1,550 to 5,200 me; X, 5,200 to 11,000 me; 
K, over li,000 me. 

c Further information on the sets discussed will be found in 
the Summary Technical Report of Division 14, Volume 2. 


8.3.3 Range and Angle 

Airborne Gun Sights [AGSJ] AN/APG-8 (inhabited 
turrets) and AN/APG-15 (remote turrets) supply not 
only range but also azimuth and elevation. The sys¬ 
tem weighs approximately 150 lb. It has the same 
range accuracy and angular coverage as does APG-5, 
but employs the conical scan described above. How¬ 
ever, tracking in angle is not as accurate as is visual 
tracking. For tail warning and defense, point-blank 
fire is used. The presentation is a dot on a small 
cathode-ray tube (G scope), which must be kept 
centered by the gunner. A damping circuit is em¬ 
ployed to minimize “jiggle .’’ Precisely as in the case 
of lead computing sights, a lag in presentation is a 
penalty. With a time constant of Ko sec (which is 
considered short) this lag is of the order of sec. 

8.3.4 Airborne Gun Laying 

Airborne Gun Laying [AGL] sets AN/APG-1, 
AN/APG-2, AN/APG-3, and AN/APG-16 are de¬ 
signed for fully automatic ranging and tracking in 
azimuth and in elevation. APG-1 and APG-2 weigh 
about 600 lb and operate in the S band. APG-3 and 
APG-16 weigh about 250 lb and are well up in the 
X band. Little is available in the current literature 
concerning tracking errors in angle. d Estimates of ac¬ 
curacy have been as optimistic as 5 milliradians but 
this depends somewhat on the angular rate. APG-16 
is engineered for use with the Sperry S-8 sight. The 
antenna is a conical scanner 9 in. in diameter 
mounted in a ball 18 in. in diameter which is fastened 
rigidly to the outside of the turret 106 (e.g., the Sperry 
A-17 ball turret in the tail of the B-32). The ball con¬ 
tains motors and gears which spin and direct the an¬ 
tenna relative to the ball. When tracking, the axis of 
the conical scan points directly at the target. The 
motion of this axis is the vector sum of turret motion 
and motion with respect to the ball. It follows that 
the relative displacement of the antenna within the 
ball is the lead, since the guns are boresighted with 
the turret, and so is an output of the S-8 system. 

8.3.5 Instability of a Radar-Gyro- 

Sight System 

In illustration of new difficulties that may arise 
when radar is used in conjunction with a sighting 
system, consider the stability problem 197 - 200 in the 

d The Armament Laboratory of Wright Field is conducting 
tests on this point. 


CONFIDENTIAL 





114 


NEW DEVELOPMENTS 


British Automatic Gun Laying Turrets [AGLT] 
Mark I. (With a director system such behavior as 
that discussed below does not occur.) The direction 
of the target is shown to the gunner as a collimated 
scope spot. A (single) gyro gunsight is used to predict 
the lead, and the guns are to be moved so as to keep 
the moving reticle of the sight on the synthetic radar 
target. Range is fed automatically. The radar has, 
however, an appreciable lag (stickiness) and the 
synthetic target is governed by the equation 

(sp + l)(y - p) = 7 - T , 

where s is a damping constant, p is the operator d/dt, 
and 7, p, r are respectively the angles that the gun 
line, radar target line, and actual target line make 
with some fixed reference line. The equation of the 
sight is familiar from Chapter 5. Neglecting ballistics, 
it is 

[(1 + &)t m p + l](y o’) = t m pyy 

where a is the angle made by the sight line with the 
reference line. 

Suppose now that the target is initially astern and 
that the guns suddenly start to move (under some 
aiming disturbance) according to y = y 0 £, where y 0 is 
a constant. If the gunner does nothing and the target 
remains astern, then for unit gun motion and 
s = 0.25, a = 0.43, t m — 1, the resulting angular 
motions are shown in Figure 5. The gunner, since the 



0 0.2 0.4 0.6 0.8 1.0 

TIME (SECONDS) 

Figure 5. Radar and sight responses: AGLT Mark 
I. (Figure from British source: Armament Depart¬ 
ment, R.A.E.) 

sight seems to be falling behind the target, will 
accelerate the guns in the wrong direction. This 
drags along the radar curve — which has the same 
initial slope as the gun line — and so aggravates the 
situation. If the gunner could actually keep his sight 


line on the radar target perfectly, then p = <r, and 
the two equations above yield, for constant t m , 

l-t m sp 2 + atmP + l]y = [(1 + a)t m p + 1>. 

The transient term is damped and the system is 
stable only if the roots of 

— t m sx 2 + at m x + 1 = 0 

have negative real parts. This means that t m s must 
be negative, which is not possible. One could obtain 
neutrality by making s negligibly small but this is not 
possible since radar spot jitter must be damped. 

It is clear that the difficulty arises since the initial 
slope of the radar curve is equal to that of the gun 
curve and the initial slope of the sight curve is only 
a/(l + a) times that of the gun curve. It seems 
plausible to decrease the slope of the radar curve by 
first reflecting the target spot from the mirror of 
another gyro unit. 200 If the equation of the auxiliary 
gyro is combined with the two original equations it 
is readily seen that a parameter can be chosen to 
assure stability of the resulting third order differen¬ 
tial equation. 

It is also possible 197 to reduce the slope of the 
radar curve below that of the sight curve by an 
electric circuit such as that shown in Figure 6, in 



Figure 6. Stabilizing circuit: AGLT Mark I. (Figure 
from British source: Armament Department, R.A.E.) 


which the input voltage is proportional to displace¬ 
ment of the gun relative to the platform. The stabiliz¬ 
ing signal is 

O+QpKl+Rp) (t ~ °° 
where a is the directional angle of the gun platform 


CONFIDENTIAL 

























NEW CONCEPTS 


115 


relative to the datum line, and P, Q, and R are cir¬ 
cuit constants. The position of the radar target is 
given by 

( _ spy r Pp 2 (y - a) . 

P 1 + sp 1 + sp (1+Qp)(l+Rp) 

If this equation is combined with the sight equation, 
p being placed equal to o, a fourth order differential 
equation arises whose stability can be established. 

8.4 NEW CONCEPTS 

8.4.1 Sight Parameter as a Function 
of Time 

One of the early proposals for improving the per¬ 
formance of lead computing sights (angular rate by 
time-of-flight multiplier) suggested that the sight 
parameter, a, be varied as a function of present 
range, r. 107 This work assumed that the target fol¬ 
lowed a straight line course and that the time-of- 
flight multiplier, t m , was the time of flight over 
present range, t p . (This antedated the calibration 
concept.) According to Section 5.3.4, “Delay in Lead 
Computation,” for positive a, the lag in lead compu¬ 
tation of the smoothing circuit may just balance the 
error in t p co if a is chosen properly. An analysis up to 
first order terms shows that a must be given by the 
expression 

r' t p r' — r 

a = — - - -; 

r + c 2 t p r — r 

where the constant c is to be chosen somewhat larger 
than the expected target speed, and r' = dr/dt p . 

From the point of view of present day airborne 
fire control this refinement has little weight for three 
reasons: (1) against curved courses approximating 
pursuit curves t p oi is in error by about 10 per cent, a 
cannot be chosen to affect this materially, and the 
calibration concept effectively does the same job 
without the necessity of awkward mechanizations of 
variable a; (2) present range input is too rough to 
be efficient in determining r'; and (3) any improve¬ 
ment in accuracy is completely masked by much 
larger operational errors. 

It is evident that such circuit modifications should 
await a renewed study of the fire control problem 
with all elements in proper focus. At present, the idea 
is much more pertinent to antiaircraft fire. 


8.4.2 Exponential Spot Sights 

A second early proposal 10 ^ 132 for improving lead 
computing was much bolder iA concept. The equation 
connecting sight angle and gun angle is 

(1 + a)t m o o = y + at m y. 

Suppose that the reticle pattern consists of a large 
number of spots which are not fixed but which have 
relative motion with respect to the reticle center. 
Let o n be the angle between a line of sight to one of 
these moving spots and the datum line. Hold this 
spot on target. Now o n will satisfy the equation 

(1 a) t m o n “h On = y -f - &t m y 

if 

o n o — K n e , 

which means that the spot must approach the reticle 
center exponentially. Continuously rotating reticle 
discs which when superposed will give spots behaving 
in this way are shown in Figure 7. 

Suppose now that the guns are slewed rapidly 
around to get on a target moving at a constant rate 
To- With the usual sight the reticle would be left 
behind and it would take an appreciable time for the 
gunner to get on target. The whole point of this 
device is to eliminate this waste time. To illustrate 
this, let the guns be in correct lead position and mov¬ 
ing at the correct rate, but suppose that the reticle 
center has not yet come in correct lead position. 
There will be, however, a spot centered on the target 
which seems fixed and toward which spots on the 
right and on the left are converging. Analytically, let 

7 = To it + t m ), o — o o and r = 0 when t = 0. 
Then, for constant t m , 

o = y Q t — ooe~ t/< ' l+a)tm 
<?n = To t + (fin — CTo)e“ </(1+a) * m . 

Consequently the spot n* for which k n * = o 0 is (and 
stays) in correct lead position since y o n * = To tm- 
Furthermore, since 

On &n* — \kn O’oJO , 

d(o n ~ On*) <0 k n > Oq 

dt > 0 k n < Oq, 

the remaining spots converge on n* as noted. This is 
not the whole story (t m was held constant and the 
field may not be wide enough) but it does demon¬ 
strate the value of the idea. It is not clear, however, 
what method would be used to get a framing circle 
to move suitably. This ingenious idea may be 


CONFIDENTIAL 








116 


NEW DEVELOPMENTS 


assessed, perhaps, as a patching-up of existing gear. 
But the principle of the rotating reticle may well be 
kept in mind. 



Figure 7. Exponential spot reticle. 


8.4.3 Aided Tracking on the Line 
of Sight 

A frequently neglected point in the design of equip¬ 
ment is ease of control (which affects the efficiency 
of operation). This phase of the design involves the 
experimental study of such details as inertia, friction, 
and cramping, using gunner subjects. It may involve 
more subtle matters. As an example of the latter, it 
has been proposed that aided tracking on the line of 
sight be used in conjunction with a single gyro 
sight. 109 - 159 


Suppose that the voltage derived from the handle¬ 
bar position is fed not only to the turret motors, but 
also to the sight after it has been differentiated (see 
footnote to Section 8.4.4). Superimpose on the volt- 
tage source for the trail coil circuits (Section 5.5.5, 
“Electromagnetic System”) this additional voltage n- 
Then the magnetic center toward which the gyro 
axis precesses is shifted through an angle 

0 = kg,. 

The gyro precessional rate is now given by 
(1 + a)tj = 7 - e + 0. 

Since 6 = y — (1 + a)\, it follows that 

tmy = tm{ 1 + o)X + X + - - • 

1 + a 


Next, if the yaw of the gun mount is a, then for a 
velocity controlled turret the absolute gun position 
7 is determined by 

y — a = B/i, 


where n is the displacement of the handlebar con¬ 
trols. 6 Using this relation to eliminate y, the lead 
equation is 

t m ( 1 + a)X + X = t m a — ~—y-1- t m Bn. 

1 + a 


The sight line equation is, putting p — d/dt, 
Pm(l + 0)V + = \jltmV + 1 ~\oL + + atm^ 


Bn 


+ 


k 


1 ~b a 


p\i. 


(The term involving da/dt can be construed as an 
error since, because of it, motions of the gun mount 
interfere somewhat with the correct positioning of 
the sight line.) Neglecting aircraft motion , give a 
sudden jerk An to the handlebars. The response Act 
of the sight line is inferred from the preceding equa¬ 
tion by supposing that the jerk is an instantaneous 
matter so that pa and pn are infinite. Hence divide 
by p and then let p —» °°. The result is 


A a 


k 

-A u. 

tm( 1 + O') 2 


This shows the direct (or positional) tracking effect 
of the coupling of handlebars to sight. On the other 
hand, the sudden change in angular velocity of the 
sight line, A (pa), caused by a sudden change in 
handlebar position dn can be obtained by solving for 


e The constant B is not always a constant for all turrets. It 
may be, for example, B + A \ n |. 


CONFIDENTIAL 






















NEW CONCEPTS 


117 


pa and neglecting Bn/p and kpi u/(l + a), i.e., the 
integral of Bn, which is Bn/p, is taken over an in¬ 
terval of zero length, and the derivative of n is zero 
at both the beginning and end of the step-jump in n- 
Hence 


A(pa) = 


a 

1 -j - CL 


A (Bn) - 


k 

tm{ 1 + «) 3 


Am. 


This shows the velocity tracking effect. For constant 
B, if 

k > Ba( 1 + a) 2 tm, 


which could happen for a too large value of k or a too 
small value of t m , the velocity response will be nega¬ 
tive and poor tracking will undoubtedly result. 

It is recommended 139 that circuits such as these be 
studied with the aid of simulating electronic circuits 
in which various parameters may be varied. 1 


8.4.4 Correction Mechanisms for 
Target Course Curvature 

A constantly recurring motif throughout previous 
chapters has been the serious effect on the deflection 
put out by computing sights attributable to the 
neglect of course curvature. Mechanisms have been 
proposed 133 which will take this into account, not 
by the naive expedient of flipping a switch from 100 
to 90 per cent of present time of flight in going from 
rectilinear to curved target paths. 

By equation (11) of Chapter 2 the correct kine¬ 
matic deflection is given quite closely in terms of the 
angular momentum, M = r 2 co, of a target of unit 
mass. This deflection may be written 

,+SMl. 

v 0 L 2 dt J 

The classical lead computing sight contents itself 
with 

A AT" 

A k = — • 

Vq 

Even this value might well be corrected by utilizing 
the rate of change of present range, assuming good 
input of range. The point of the present discussion, 
however, is to indicate a mechanism which can be 
combined with an ordinary gyro sight to yield the 
angular momentum correction. 

Use the single gyro from a K-15 sight, without the 


f Such studies were made, for example, at the Marcellus 
Hartley Laboratory of Columbia University under J. B. 
Russell. 


ballistic coils. The deflection of the mirror of this 
gyro is made proportional to M by feeding a current 
to the range coils through a nonlinear potentiometer 
(r 2 ) whose position is determined by present range r. 
The light reflected from this gyro mirror is to come 
through a reticle of such variable transparency that 
the intensity of passing light at any point is propor¬ 
tional to the logarithm of the distance from the 
center. After reflection from the gyro mirror the light 
hits an opaque screen with a small central hole be¬ 
hind which is a photocell. Since only that part of the 
reticle yielding light proportional to In M hits the 
photocell, the voltage output of the cell is propor¬ 
tional to In M. This voltage is amplified linearly and 
fed into a circuit containing a condenser, which 
differentiates In M, g and containing a range poten¬ 
tiometer which introduces t f / 2. By means of a servo, 
a rotation proportional to ln[l + (£//2)(dln M/dt)~\ 
can be introduced in one side of a differential. The 
input to the other side is proportional to In r. The 
differential’s output (the sum) may then be sent to 
the range potentiometer of the usual gyro sight, and 
the correction has been effected. 

The gain in accuracy is most significant when this 
correction is made. The effect of roughness in input 
of r and co is, nevertheless, still open to question. 

It is also possible to utilize variable speed drive 
computers (ball-cage integrators) to apply this cor¬ 
rection to the stabilized S-3 sight. 134 

8.4.5 An Own-Speed Plus Rate 
Mechanization 

Lead computing sights have decomposed deflec¬ 
tion into kinematic and ballistic components. Other 
decompositions and consequent mechanizations are 
possible. 23 To keep one such new idea clear, various 
parts of the proposal (corrections for target and gun 
platform accelerations, and use of range rate) will 
not be considered. 

A basic lead equation of gunnery is equation (1) of 
Chapter 2. It is 

. . Vg . vt . 

sm A = — sin r — q — sm a = A 0 — At . 

Vo Vo 

The mechanization suggested uses a standard own- 
speed sight (the 0 computer) to mechanize v G sin 

g If the input voltage is in series with a small resistance and 
a condenser of large impedance, the voltage drop across the 
resistance is approximately RCdE/dt. This small voltage may 
be fed to an amplifier. 


CONFIDENTIAL 








118 


NEW DEVELOPMENTS 


t/v o with no percentage correction. It uses a rate 
computer to deal with v T sin a/v 0 (the T computer). 
Since target speed and approach angle are not 
known, the T computer proceeds in what seems to be 
a circular fashion by accepting as its input, in dif¬ 
ference, vg sin t/v o, and ru/v 0 (as determined by 
ranging and gyro-tracking); and yielding, through a 
smoothing circuit 

a(r)\ T + \t = Vg sin t/v 0 — ru/v 0 , 

an angle Xy. (The smoothing circuit selected suggests 
that a be a function of r to give less damping at 
greater ranges for which vt sin a/v 0 is relatively con¬ 
stant.) This output \t must be combined with the 
ballistic factor q. For projectiles of proper shape and 
in that velocity range for which the three-halves 
power drag law holds, 


1 



where p is the relative ballistic air density, and 

b — 0.00323 (when the unit of length is yards), 
b = 0.00186 (when the unit of length is feet). 

The actual air range covered by the bullet can, in the 
absence of useful range rate, be approximated by 

P = cr, 

where c is a constant. For approach angles that do 
not exceed 30°, this is very effective. The velocity of 
departure, u 0 , is given adequately by 

V5_VS(. + ^), 

where ^Gproj is the component of mount velocity 
along the present range line. 

It is the above quantity q that is to be computed 
with precision. Through it, i.e., through this decom¬ 
position, ballistics are automatically taken into ac¬ 
count, and no separate trail ballistic circuits are 
needed. (The time of flight, t f , which is normally de¬ 
termined by lead computing sights, will enter here 
only in making the small curvature correction. See 
equation (7) of Chapter 2. The simple estimate 
tf = r/v o should prove satisfactory.) 

This own-speed rate system has real advantages 
when used against curves approximating pursuit 
curves flown against one’s own bomber or against a 
neighboring bomber in the formation. The reason is 
that only a small part of the total lead is subjected to 
the smoothing circuit. Consequently, percentage 


errors of the same size as those of a normal rate-time 
sight now generate a small absolute error. Against 
forward hemisphere attacks, the advantage is even 
more marked. The initial error will be small and at 
close ranges when the required deflection builds up 
rapidly — primarily because of the rapid change in 
own-speed deflection — gun pointing should be quite 
good in virtue of the presence of the 0 computer. 


8.4.6 General Approach to the 
Tracking Equation 


The study of a complete fire control system de¬ 
mands a knowledge of the tracking equation which 
connects handlebar motion to sight line motion. In 
the case of manual operation by a human gunner a 
theoretical description of the complete system is 
difficult since the gunner’s responses cannot be 
described exactly — he is an imperfect servo¬ 
mechanism. Consequently, up to the present, resort 
has been made to experimental and statistical pro¬ 
cedures rather than theoretical analysis. It is evident 
that the advent of automatic inputs will render 
mathematical methods more attractive. But the 
elements of a general theory are useful even in the 
case of manual operation 53 for which only probable 
behavior on the part of the operator may be assumed. 

To illustrate the methods of the theory, consider 
aided tracking combined with a lead computing sight. 
The gun postion, 7, is connected to the handlebar 
angle, /x, by 

7 = Ap + Bu. 

The sight equation relates gun angle to the sight line 
angle, <r, by 

atm 7 + 7 = (1 + a)t m a + a. 

If 7 is eliminated between these two equations, then 

(1 + a)t m a + [1 + (1 + a)im](T = Aat m u + 

[A(l + atm) + Batman + 2?[1 + ai rt /\ii. 
It will be noted that t m is permitted to vary with 
time. The solution of this equation may be written 
in the form 53 


where 

K(t,s ) = B + 


-r«,«)/l+a 


r a b -] _ r 

|_ (1 + a) 2 t m (s) 1 + a J 
The artificial time, r, moves faster than t if t m < 1. 


CONFIDENTIAL 









NEW TACTICS 


119 


More generally, suppose that the gun-handlebar 
equation is 

Ry = R'n, 

and that the gunsight equation is 

Py = Qv, 

where R, R', P, and Q are polynomials in p — d/dt 
whose coefficients depend on t m . If t m is held constant, 
the tracking equation is 

RQa = PR' )jl. 

The tracking equation with t m variable is not ob¬ 
tained so readily. Since, however, the change in t m 
due to ranging has but little effect on the nature of 
the tracking, this equation may be used to infer 
various properties of the tracking performance. For 
example, there are various solutions of the tracking 
equation depending on the initial handlebar motion. 
If the difference of some pair of these solutions does 
not approach zero, the tracking is essentially unstable. 
(The operator could track a target at rest by handle¬ 
bar motion not approaching a position at rest.) The 
difference between any pair of solutions must satisfy 
the equation 

PR'y = 0. 

This has a solution not approaching zero if and only 
if PR' has a nonnegative root. (For example, for 
aided tracking with a lead computing sight, the roots 
of PR' are —B/A and — l/at m - Stability is the 
normal situation.) The general equation can also be 
used to study hunting and the response to handlebar 
jerks. 

The most useful form, here, of the solution of the 
tracking equation is given by operational methods 
employing the convolution integral. 53 The reason is 
that the resulting solution expresses cr(t) as a 
weighted average of earlier values of y (t). This was 
illustrated explicitly in the preceding paragraph. The 
solution of the tracking equation with constant co¬ 
efficients is 

a(t) = Cy(t) + f K(t - t')y{t')dt' (t > to) . 

J to 

Further details and interpretation will not be 
given. In summary, the point to be made is that in 
future problems of overall system design and system 
response, methods of this nature will be exploited. 
These general methods will doubtless require exten¬ 
sion to nonlinear systems, only special examples of 
which have been discussed. 12 


8.5 NEW TACTICS 

8.5.1 Offset Guns in Conventional 

Fighters 

The conventional fighter can mount a heavy for¬ 
ward firing battery at minimum cost in installation 
weight and balance. The penalty is that the entire 
aircraft must be aimed and must fly a predictable 
course, the pursuit curve, if a succession of hits are 
to be scored on a bomber target. At the other end of 
of the scale is a turreted fighter which may fire as 
flexibly as its target. The P-61 is an example. This 
type is penalized not only by increased weight and 
size, and crew, but also by all the difficulties in taking 
deflection experienced by the defending bomber. A 
middle course appeared feasible during the closing 
months of World War II. If fixed guns are installed 
at a large angle to the thrust axis, installational 
weight is not significantly increased, the modification 
is possible for many existing fighters, and there is no 
increase in crew. The following questions must be 
answered. (1) What are the advantages? (2) What is 
the aiming problem with an offset installation? 
(3) What are the disadvantages? Considerable study 
has been devoted to these points. 42 ’ 125 ’ 159 ’ 164 ’ 173 

8.5.2 Attack by Pacing Behind 

and Below 

An attack is advantageous to a fighter if his 
attack path is nearly rectilinear, so that coordinated 
banked flying under conditions of high load is not 
required, if the deflection required is small and does 
not change rapidly, and if continuous fire may be put 
into the bomber until destruction ensues. 

The simplest method of gaining these advantages 
with an offset installation is to mount the guns point¬ 
ing upward and forward in a vertical plane at such 
an angle that, for pacing flight below and behind, 
and over a considerable range interval, the backward 
curve of the trajectory caused by trail is sensibly 
balanced by the forward and downward curve at¬ 
tributed to gravity and by the parallax correction 
caused by separation of guns and sight. The feasi¬ 
bility of this depends on the ballistics of the ammu¬ 
nition employed. For example, 42 using API M8 
caliber .50, an offset angle of 30°, and a parallax 
distance of 2 yd, the total lead required is: 4 milli- 
radians (±1) for a speed of 175 mph, 6 milliradians 
(+ 1) for a speed of 250 mph, 8 milliradians (±2) for 


CONFIDENTIAL 



120 


NEW DEVELOPMENTS 


a speed of 325 mph. These negligible variations cor¬ 
respond to an interval in range of from 100 to 400 yd. 
Given an offset angle and a range interval it is 
necessary to design special ammunition to achieve 
constancy of lead. Then the attacker need not de¬ 
termine speed and range with precision. With un¬ 
suitable ammunition, for example, typical Japanese 
20-mm ammunition (y 0 = 2,358, c 5 = 0.240) fired at 
an offset of 45° with a parallax of 2 yd, the required 
deflection increases from 13 to 28 milliradians (at 250 
mph) as the fixed range at which the attack is made 
changes from 200 to 500 yd, and from 18 to 42 milli¬ 
radians (at 325 mph) over the same range interval. 
The attack is still quite possible, but if it is not made 
at a precalculated speed and range, the pilot must lay 
off deflection. 

8.5.3 Attack on Collision Course 

A second possibility is to attack on a collision 
course. The attacker crabs in on a path which in 
relative motion is a straight line terminating at the 
bomber. If the guns are offset so as to lie slightly 
forward of this relative path (to allow for trail) then 
continuous fire is possible with but little change in 
the required small deflection. (During World War II 
the German Air Force occasionally tried skidding 
attacks with conventional fighters. h Evidently this 
is equivalent to mounting guns at a small offset. 
Without the aid of a gyroscopic lead computing sight, 
it is difficult to fly.) Using the notation of Figure 1 
of Chapter 3 and formula (1) of Chapter 2, the (lead) 
angle between the relative path and the bore axis 
must be 1000^ sin 8/v 0i where v F sin 8 = v B sin 0 is 
the collision course condition. The term l = q — 1 is 
an almost linear function of pP/c 5 . Consequently, 
the larger the ballistic coefficient is, the smaller the 
Spread in l over a given range interval will be, and the 
more efficient a fixed average offset from the relative 
path will be. 

Two methods of making this attack are possible. 
Suppose first that the sight is offset by an angle 8 and 
the guns by an angle 8 — lv F sin 8/v 0 radians where l 
is an appropriate average. Use a fixed fighter speed 
and fly so as to keep the target centered in the offset 
sight. Then for a value of 0 different from the opti¬ 
mum given by v F sin 8 = v B sin 0, the fighter’s path 
will curve (quite gently) and certain aiming errors 


h Interrogation of Th. W. Schmidt at Coburg, Germany. 
E. W. Paxson, June 5-8, 1945. 


will occur. As a numerical example, 164 if v F = 440 
fps, v B = 321 fps, altitude = 20,000 ft, and if API 
M8 ammunition is used, then, with sight offset 
45° 15' and gun offset 45°, the angle off can vary 
from 70° to 120° off nose, and the aiming error will 
vary only from a maximum of 9.7 milliradians be¬ 
hind a (point) target to a maximum of 5.8 milli¬ 
radians ahead of it. As a second method, suppose that 
the pilot keeps the target centered in his offset sight 
but adjusts his speed according to v F = v B sin 0/sin 
5. In this case a true rectilinear course is flown and 
the error will again be slight. 

Tactically, the accuracy of the fighter’s fire should 
not depend critically on initial range and bearing. 
Allowance for error must be permitted. Similarly, a 
variation in the calculated speeds of fighter or target 
must not cause the accuracy to deteriorate too much. 

8.5.4 Discussion of Offset Gun 

Attacks 

The discussion above indicated that specialized at¬ 
tacks of this nature are possible. But there are serious 
disadvantages. If a fighter armed with offset guns 
attempts to make anything like a pursuit curve at¬ 
tack in a slant plane, his required motion will be a 
twisted space curve with curious torsion, which prob¬ 
ably cannot be flown. In particular, a fighter without 
a forward battery can hardly attack another fighter. 
Even against a bomber, the arguments of Sections 
8.5.2 and 8.5.3 are predicated on a bomber that main¬ 
tains a straight and level course. Rather mild evasive 
action can cause the attack to abort. Finally, assum¬ 
ing the bomber does cooperate by flying a rectilinear 
course, the fighter is an equally easy shot for any of 
the fire control systems of Chapter 5 which have well 
adjusted, i.e., variable, trail allowances. 

In view of these disadvantages, it appears that 
these attacks are most promising when made by a 
night fighter. For this case, the parallel course attack 
of Section 8.5.2 is probably best. The defensive 
radar tail-cone search can be avoided, and, if pro¬ 
vision is made for both upward and downward 
firing guns, 176 - 189 proper advantage of night sky con¬ 
ditions may be taken. These attacks would also be 
dangerous to a bomber which has stripped its arma¬ 
ment to tail-cone defense to obtain speed and addi¬ 
tional pay load, if it is assumed that in so doing the 
fighters could only make tail-cone attacks for reasons 
of load and buffeting. It may also be kept in mind 
that the paths of Section 8.5.3 are collision courses 


CONFIDENTIAL 




SUMMARY 


121 


and so may be as attractive to a guided or self-guided 
missile as is a homing course. 

8.5.5 Upward Barrage Fire 

Turning from conventional machine gun armament 
in fighters, upward barrage fire at close range has 
been proposed. 1 The installation consists of 40 
barrels, each containing but one round, mounted to 
fire sensibly upward. The attack is to be from the 
front and initiated by a dive down and across the 
bomber’s bow. Upon pull-up below in the same 
vertical plane the tubes are ripple-fired automatically 
by acoustic pickup from the bomber’s propellers or by 
optical pickup of the bomber’s shadow. The range of 
fire is to be 50 to 100 m, which means that the violent 
motion described on the attacker’s part is necessary 
if he is to arrive unscathed at the firing point. Reli¬ 
ance is placed on a linear pattern approximately 80 
meters long, rather than on a precise estimate of de¬ 
flection. 

There are certain technical difficulties. To avoid 
high recoil a low muzzle velocity of the order of the 
firing aircraft’s speed is to be used. Consequently, 
large caliber (2.0 to 2.5 cm), high-explosive, and 
fused projectiles must be used. The initial yaw is of 
the order of 40°. For a spin stabilized projectile 
(rifling of 30 calibers per revolution) this yaw does 
not damp sufficiently to assure an angle of impact 
favorable to the functioning of the fuse. A fin stabi¬ 
lized projectile is more satisfactory. 214 

8.6 SUMMARY 

Section 8.1 limits the chapter to those ideas in 
hand during the closing months of World War II. 
The next phase in fire control development should be 


* Interrogation of Doetsch, Blank, Rossman, Schiissler, 
Hackeman, at the Luftfahrtforschungsanstalt, Braunschweig, 
Germany. E. W. Paxson, June 28/29, 1945. 


a judicious blend of principles now known to be 
essential. The plea on behalf of the designer is made 
that the shape of expected tactical situations be 
made available. 

Section 8.2 deals with systems that stabilize the 
gunner’s line of sight by removing perturbations 
caused by aircraft motion. The theory of the rate- 
stabilized Fairchild S-4 system is brought to the point 
of deducing the complete circuit equations. The 
position-stabilized Sperry S-8B sight is discussed 
similarly although the circuit response equation is 
obtained experimentally rather than analytically. 
Stabilization does not extend to aircraft accelera¬ 
tions, and the time constants of follow-up circuits 
must be carefully considered. 

Section 8.3 discusses the use of radar in supplying 
automatic inputs, in range and in bearing, to a fire 
control computer. Conditions of instability in track¬ 
ing are mentioned briefly. 

Section 8.4 lists as new concepts (1) the variation 
in the sight parameter as a function of range, (2) a 
mechanism designed to cheat partially the transient 
time in a lead computing sight, (3) the use of aided 
tracking on the line of sight, (4) an ancillary circuit 
for lead computing sights which gives a correction 
for target course curvature through the rate of change 
of the angular momentum of the target about the 
gun, (5) the mechanization of a decomposition of 
lead into own-speed allowance and target motion 
allowance, rather than into kinematic and ballistic 
deflections, and (6) the elements of a general theory 
of system response to control movements. 

Section 8.5 considers as new tactics (1) the use of 
a fixed offset gun in a fighter attacking on a pacing 
course, (2) the use of such an installation on a col¬ 
lision course attack, and (3) upward barrage fire at 
close range. 

No account was given of air-to-air rocketry or 
air-to-air bombing. One must also look elsewhere for 
information on guided and semi-guided missiles. 


CONFIDENTIAL 








PART II 


ROCKETRY 


CONFIDENTIAL 





Chapter 9 


FIRE CONTROL FOR AIRBORNE ROCKETS 


9.1 INTRODUCTION 

I n the spring of 1944, the program of the Applied 
Mathematics Panel in rocketry was initiated by a 
request from Division 7, NDRC, for AMP’s cooper¬ 
ation in the development of the best possible comput¬ 
ing sights for rocket firing, where the term best was 
to be interpreted as applying to practicability and 
availability in point of time, as well as to accuracy. 
There were subsequent requests from Division 7 and 
from the Navy for assistance with various aspects of 
the rocket sighting problem. As a result of these re¬ 
quests AMP’s activities in this field fell into three 
categories. 

First, there was a general study of what sighting 
methods are feasible. For rockets, the essential prob¬ 
lems involved in this question have to do with bal¬ 
listic formulas, 13,17 attack angle and skid, 14 - 16 the 
effect of wind and target motion, how these various 
factors affect each proposed sighting method, and 
how tracking affects and is affected by them. 7 ’ 20 Two 
AMP reports 1,2 consider general aspects of the 
problem, give formulas relating to the quantities in¬ 
volved, and study in some detail various sighting 
methods. This chapter is based largely on the ma¬ 
terial in the first of these reports 1 which is considered 
fundamental to any future work concerned with the 
development of computing sights for rocket firing. A 
brief discussion is presented here of the principal 
ballistic questions involved in the problem; a simpli¬ 
fied treatment of kinematic lead is given; and certain 
of the questions involved in the design of a com¬ 
puting mechanism are set forth. 

Second, several specific proposals for rocket sights 
were made and studied. 2 - 10 For example, a range¬ 
finding method using the Mark 23 gyro sight was 
studied; 12 a miniature rocket sight called PARS, 
consisting of a mechanical addition to an altimeter 
with an electrical connection to a sight head, was 
proposed 8 ’ 10 and was scheduled for test. A brief 
description of the characteristics of this sight is given 


in the Division 7 Summary Technical Report. 4 
AMP also participated in the program for the de¬ 
velopment of a 'pilot's universal sighting system 
[PUSS], 11 - 15 ’ 29-31 reported in Division 7 Summary 
Technical Report; and made related studies of toss 
rocketry 18>23 and of toss bombing in the presence of 
wind. 21 ’ 22 ’ 25 - 27 

Third, a method was formulated for determining 
whether pilots who are attempting to fly a diving 
course at a target skid badly enough to cause a sub¬ 
stantial error in rocket fire. 3,22 For further informa¬ 
tion concerning this and other detailed questions 
studied by AMP, the reader is referred to the reports 
listed in the bibliography at the end of this volume. 

One of the principal advantages of rockets over 
shells is that no heavy gun or gun mount is needed in 
rocket firing, since the recoil is taken up by the 
ejected gases. For stability, the rockets considered 
here are constructed with fins near the tail. These 
fin stabilized rockets, fired forward from aircraft, 
have become an important weapon of warfare. Be¬ 
cause of their relative newness as an aircraft weapon, 
the problem of how they may best be aimed is not 
yet fully understood, and no very satisfactory com¬ 
puting sight was in service at the termination of 
hostilities. The problem is considerably more compli¬ 
cated than that of aiming bullets. In the first place, 
due to the presence of the fins, the rocket picks up 
most of its velocity in the direction of motion of the 
airplane in the surrounding air mass; this may differ 
from any fixed direction in the airplane by a con¬ 
siderable amount. Moreover, to determine this direc¬ 
tion is a difficult job. Secondly, except at short 
ranges, the gravity drop must be accounted for; it 
can be many degrees at 3,000-to 4,000-yd range. This 
requires some (direct or indirect) knowledge of the 
range; 6 9 and there is no simple known instrument 
for use in a fighter plane which gives the range. A 
third consideration which affects the whole char¬ 
acter of the problem is the temperature of the 
propellant grain which is dependent upon the initial 


CONFIDENTIAL 


125 


126 


FIRE CONTROL FOR AIRBORNE ROCKETS 


temperature (determined by conditions of storage) 
and is affected by the altitude at which the airplane 
may have been traveling for some time before the 
attack. Finally, the projectile takes normally two to 
ten seconds to reach the target. By this time the tar¬ 
get may have moved a considerable distance relative 
to the air mass. On the other hand, since the ammuni¬ 
tion dispersion amounts to only a few mils, and since 
most targets for aircraft rockets are rather small 
— tanks, locomotives, gun emplacements, and small 
shipping, for example — it would be of great ad¬ 
vantage to obtain an accurate solution of the sighting 
problem. 

9.2 FIN STABILIZED ROCKETS 

Rockets are normally held under the wings of the 
aircraft. When fired, they accelerate, increasing their 
velocity to a maximum of usually 600 to 1,500 fps, 
depending mainly on the type of rocket. Burning 
time for most rockets is about a second; it depends 
greatly on the temperature of the propellant, being 
shorter at higher temperatures. After burning, the 
rockets slow down again, due to drag. 

The launchers commonly used now are zero length 
launchers. The rocket is suspended at two points, 
and becomes free after it has traveled an inch or less. 
Since it is already moving through the air mass with 
the velocity of the aircraft, the air acts strongly on 
the fins, turning it into the wind as soon as it leaves 
the launcher if it is not already pointed in that direc¬ 
tion. Angular momentum carries it beyond this 
point, so that its direction oscillates with decreasing 
amplitude about the direction of its vector velocity. 

For sighting purposes, one may neglect these 
oscillations, provided that one determines their over¬ 
all effect on the general direction of the path. This 
is accomplished by defining an effective launcher line, 
along which the rocket is supposed to start out. 

9.2.1 The Effective Launcher Line 

The effective launcher line is a line a certain frac¬ 
tion / of the way from the launcher line (the line of 
the rocket at firing) to the velocity vector relative to 
the air mass of the airplane firing the rocket. 

The value of / is commonly between 0.8 and 0.99. 
For the 5-in. HVAR rocket and in the vertical plane, 
/ is given approximately by the formula 

(1) 


where T is the propellant temperature in degrees 
Fahrenheit and Vi is the indicated airspeed of the 
launching airplane in yards per second. Thus / de¬ 
pends on indicated airspeed and propellant temper¬ 
ature, but not on dive angle. Numerical values of / 
are given in a report 5 which we shall hereafter de¬ 
scribe as the “tables.” 



The/factor in the lateral plane need not equal the 
factor in the vertical plane, but probably is close to it. 
Accordingly, the effect of skid (motion perpendicular 
to the plane of symmetry of the aircraft) is im¬ 
portant: Suppose, for example, that the velocity 
vector of the aircraft points 20 mils to the right of its 
plane of symmetry. Then the rocket will turn 20/ 
mils into the wind. If the target were fixed and if the 
plane of symmetry of the aircraft contained the 
target, this would lead to an error of 20/ mils, which 
is of the order of 18 mils. 

The purpose of defining an effective launcher line 
and the / factor is this: In our analysis the effective 
launcher line is taken as the initial direction of the 
trajectory. A knowledge of this effective initial direc¬ 
tion is necessary to permit the determination of the 
manner of flying the launching aircraft which will be 
most likely to achieve a hit. 

If a long launcher (rail or tube) were used, or if 
the rocket propellant were faster burning, / would be 
smaller. For a bomb,/ = 1. For a machine gun bullet, 
the initial velocity of departure is the vector resultant 
of the velocity of the airplane acting in the instan¬ 
taneous direction of motion and the velocity pro¬ 
vided by the propellant, acting along the bore axis 
(launcher line); thus / is obtainable from a vector dia¬ 
gram. 

In the case of rockets, the effective launcher line 
gives us the effective initial direction of the trajectory. 
The trajectory itself, relative to its effective initial 
direction, is then given by the angular gravity drop 
and time of flight, which we now discuss. 


CONFIDENTIAL 





AERODYNAMIC ASPECTS 


127 


9.2.2 The Angular Gravity Drop X g ; 
the Time of Flight t f 

If the rocket had its full velocity at the start, it 
would follow approximately a parabolic path, and 
the angular drop \ g due to gravity from the effective 
launcher direction to the rocket would be approxi¬ 
mately proportional to the time the rocket had 
traveled to any given point, and hence to the range r 
to that point: \ g = Bir. However, the slow speed of 


EFFECTIVE LAUNCHER 



the rocket in the first part of its trajectory lets it fall 
more per unit distance ; it also points down more, and 
hence picks up speed in a somewhat downward direc¬ 
tion. The net result is that the drop \ g is a certain 
amount B 0 more than the above amount Bir. Both 
constants B Q and Bi are approximately proportional 
to the cosine of the dive angle y from the horizontal 
to the effective launcher direction in the vertical 
plane. Accordingly, we may write 

\ g = (A 0 + Air) cos 7 in mils, (2) 

as a good approximation, where A 0 and A\ are con¬ 
stants. For the 5-in. HVAR, A 0 is about 15 mils, and 
Ai about 16 mils per thousand yards; they depend 
on the plane’s airspeed and the propellant tem¬ 
perature. 

Because of the slow speed at the start of burning, 
the time of flight tf , at fairly short ranges, is not 
proportional to the range but is nearer a constant 
plus a multiple of the range. At longer ranges, when 
the speed has died down again, the time of flight is 
approximately proportional to the range. A quadratic 
function of range with coefficients depending slightly 
on own speed gives tf to within 0.1 sec. la 

Numerical values of gravity drop \ g , angle of fall, 
and time of flight tf are given in the “tables.” 5 At 
r= 3,000 yd, for the 5-in. HVAR rocket, \ 0 may be as 
much as 100 mils; t f) as much as 7 sec. Formulas for 


these quantities in terms of range to target at impact, 
dive angle (omitted in the case of time of flight), air¬ 
speed of launching airplane and propellant temper¬ 
ature are available. 113 

9.2.3 Average and Instantaneous 

Projectile Speeds 

The average speed V is obtained by dividing the 
range r by the time of flight tf. Some values of V for 
the 5-in. HVAR, at an own speed of 300 knots and a 
propellant temperature of 70 F, are as follows. 

r 500 1,000 1,500 2,000 3,000 4,000 yd 
V 435 500 518 519 508 490 yd per sec 

Thus 500 yd per sec is a good round value. At lower 
temperatures, it takes longer for the rocket to burn 
and hence pick up its maximum speed; hence the 
flight times are longer, and the average speeds are 
less. 

A formula giving roughly the variation of V with 
the plane’s airspeed v kn in knots or v in yards per 
second is: 

V = 500 -f 0A5(v kn - 300) = 500 + 0.8(» - 169). 

The variation with propellant temperature is not 
large, and not worth taking account of, considering 
the inaccuracy already present. 

The instantaneous projectile speed , for instance at 
v = 300 knots, T = 70 F, is 600 yd per sec at 500 yd 
range, 503 yd per sec at 2,000 yd range, and 400 yd 
per sec at 4,000 yd range. 

9.2.4 Dispersion 

The dispersion of actual rockets about their aver¬ 
age position (all being fired under the same con¬ 
ditions) is only a few mils. The radius of the 50 per 
cent cone may be of the order of 7 mils. 

9.3 AERODYNAMIC ASPECTS 

It will be assumed that the launching aircraft is a 
fighter plane. Unless otherwise specified units will be 
yards, seconds, and radians. One radian is equal to 
one thousand mils. 

9.3.1 Skid, Angle of Attack, Dive 
Angle, Other Angles 

Let v* be the vector velocity relative to the air 
mass of the aircraft at an instant. We break v* into 


CONFIDENTIAL 



128 


FIRE CONTROL FOR AIRBORNE ROCKETS 


two components: v in the plane of symmetry of the 
aircraft and v s perpendicular thereto : 

v* = v + v a . 


For altitudes h less than 11,000 ft, the relation be¬ 
tween indicated airspeed Vi and true airspeed v is 

Vi/v = e~ l A& x 10 6h , 


The vector v s is the skid. We use the term true air¬ 
speed for the magnitude v of the vector v. (Since skid 
is small, v is sensibly equal to v*, the magnitude of the 
vector v*.) The skid speed is v s , the magnitude of the 
vector v s . Since v s is small compared to v , the angular 
amount of skid is v 8 /v radians. 

In the plane of symmetry of the aircraft there is a 
direction in which the rockets are held prior to their 
launching. This direction is known as the launcher 
line or datum line. (In general, the datum line is 
merely a line of reference in the plane of symmetry 
of the aircraft. We assume for simplicity that the 
launcher line, datum line, and thrust axis are the 
same. If these lines differ appreciably, the necessary 
adjustments in our analysis can and should be made.) 

The angle of attack a is the angle from the datum 
line to the aircraft’s velocity v in its plane of sym¬ 
metry. We adopt the convention in our diagrams 
that an angle in the clockwise direction is positive. 
Thus the angles a 0 , a and fa in Figure 3 are negative 
and the other indicated angles are positive. 

Consider the plane of symmetry of the aircraft, and 
suppose that it is vertical. Let 5 be the angle from the 
horizontal to the datum line. If there were no gravity 
acting, the aircraft (in steady flight) would move 
approximately along a hypothetical line in its plane 
of symmetry; this line is called the zero lift line. The 
zero lift line is at some angle a 0 from the datum line; 
being above the datum line in the figure, a 0 is nega¬ 
tive. Due to the influence of gravity, the flight direc¬ 
tion is at some angle a — a 0 below the zero lift line. 
The lifting power of the wing is approximate^ pro¬ 
portional to the square of the airspeed; it is also 
proportional to a — a 0 up to a certain point. Hence 
« — «o is inversely proportional to the square of the 
airspeed. The appropriate speed here is the effective 
airspeed v,. depending on the density of the air, 
rather than true airspeed, v. The airspeed meter feels 
Vi rather than v , so Vi is called the indicated airspeed. 
In steady horizontal flight, we may write, therefore, 

h 

a - a Q = -jg , (3) 

Vi 

where g, the acceleration due to gravity, is 10.7 yd 
per sec 2 and b is a constant of the dimension of dis¬ 
tance. For combat planes, b is commonly between 
60 and 85 yd. 


where h is measured in feet. The indicated airspeed 
is less than the true because the air pressure is re¬ 
duced by altitude. 



HORIZONTAL 
ZERO LIFT LINE 

FLIGHT DIRECTION,_y 

EFFECTIVE LAUNCHER 
LINE 

DATUM LINE * 
LAUNCHER LINE* 
THRUST DIRECTION 


Figure 3. Angles associated with rocket trajectory. 


The flight angle <f> is the angle from the horizontal 
to the line of flight. The dive angle y of the rocket is 
the angle from the horizontal to the effective launcher 
line. 

In a steady climb or dive, only the component g cos 
8 of gravity need be considered. (Perhaps some angle 
8' different from 8 should be used here; it is difficult 
to determine exactly what angle fits best, but the 
final results would probably not differ by very much.) 
If the airplane is nosing up or down, there is centrip¬ 
etal acceleration a# in the direction normal to the 
flight path to be taken into account. If R is the radius 
of curvature of the flight path, and <£ = d<t>/dt is the 
time rate of turn of the flight path, then, since 
4> = v/R, 


(In = ~ = v<f>. 
Jti 


( 4 ) 


Thus, in general, g in equation (3) should be replaced 
by g cos 8 — a Further, since the lift required on the 
airplane is proportional to its weight W, b is propor¬ 
tional to W. Hence equation (3) now becomes 

a = a Q + \ {g cos 8 — vj>) f b = b'W , (5) 

Vi 

where b' is a constant. This relation is the attack angle 
formula, the attack angle a being the angle from the 
datum line to the flight line. 

Observe that we are really interested in the effec¬ 
tive attack angle of the rocket, the angle from the 
datum line to the effective launcher line of the rocket. 
We take this to be fa, where a is given by equation 
(5) with constants a 0 and b properly determined for the 
particular plane and rocket by experimental firings. 


CONFIDENTIAL 




AERODYNAMIC ASPECTS 


129 


Typical values for a fighter plane are a 0 = —40 mils 
and b = 75 yd. 

We may rewrite equation (5) by using the rela¬ 
tion a = 0 — 8 which is clear from Figure 3. Then 
we obtain 


where 


bv . _ 

10 + 0 — 5 + a , 

(6) 

bg cos 5 

a — a 0 + 2 

(7) 


The time constant 

_ bv 
T = — 2 sec 

A 

of equation (6) is of the order of 0.5 sec, since b ^ 75 
yd and v ~ Vi ~ 150 yd per sec. 


Thr = 1,830 lb, v = 345 mph = 300 knots = 507 
ft per sec, p = 0.0021 slugs per ft 3 ; these give, using 
g = 32.2 ft per sec 2 , 

230,000 


r = 


= 6.72 sec. 


1,830 + 32,400 
Thus the flight direction approaches the thrust direc¬ 
tion very slowly; it would take 7 sec for it to go six- 
tenths of the way from any initial direction, assuming 
the thrust direction held fixed. Stated in other words, 
any skid the plane may have disappears very slowly , 
unless it is eliminated by changing the bank to bring 
the flight path around. 

The lateral motion of the aircraft when there is 
bank but no skid has been studied lc but will not be 
considered here. 


9.3.2 Response of Flight Angle <0 
to a New Fixed Direction 
of Datum Line 

Suppose that the pilot pulls the nose of the air¬ 
craft up or down until a fixed sight is on the target 
and then holds it there. He will then fly so as to keep 
5 constant after a certain instant. In this case equa¬ 
tion (6) implies that the flight path tends toward the 
final direction 

<f> = 8 4 “ OL 

and moves six-tenths of the way from its initial to 
final direction in T sec, since e~ l = 0.4 approximately. 

If the plane is banked, the attack angle formula 
equation (5) may be applied in the plane of sym¬ 
metry of the aircraft, except that g cos 8 must be 
replaced by g cos 8 cos /?, if is the angle of bank 
(measured about the thrust axis). 

Now consider motion in the lateral plane, first 
assuming the plane is not banked but is perhaps 
skidding. If the flight line is not in the plane of sym¬ 
metry, there will be a cross-wind force, tending to 
bring it back. If 0 and 8 now denote angles in the 
lateral plane from a fixed direction to the flight 
direction and to the datum line, respectively, it may 
be shown that 

T'<j> + 0=5, (8) 

the time constant T' in this case being, at least for 
motion in a horizontal plane without banking, 

Wv/g 

rjv = -- . (Q\ 

Thr + 0.2pv 2 S K J 

Here Thr is the thrust, p the air density, and S the 
wing area. For the P-47D, for example, at 5,000 ft 
altitude, we may take W = 14,600 lb, S = 300 ft 2 , 


9.3.3 Tracking with a Fixed Sight 

The problem of tracking may be looked at from 
two points of view: (1) How well can the pilot 
keep the sight line on the target ? (2) How well can 
he keep the flight direction (or better, the effective 
launcher direction) where required? If a perfect com¬ 
puting sight were used, solving (1) would solve (2). 
At the present time, sights do not correct for skid, 
though they may correct for most of the attack angle; 
it is therefore necessary for the pilot, as far as pos¬ 
sible, to avoid skidding. In many cases, wind and 
target motion are not corrected for; the pilot may 
make partial correction by pointing his sight at some 
other point than the target. This problem is not one 
of tracking. A computing sight may have a sight line 
which moves relative to the airplane in a manner de¬ 
pendent on various inputs, in particular, on the 
manner in which the plane is being flown. The prob¬ 
lem of tracking can then only be studied when this 
sight line motion is known. Here, we shall consider 
only the problem of holding a fixed sight on the tar¬ 
get, and discuss briefly the effect on the flight path. 

Suppose first that the fixed sight is approximately 
along the thrust axis. Consider what motion of the 
sight line will result from various manipulations of 
the airplane controls. Moving the stick back or for¬ 
ward will raise or lower the nose, thus raising or 
lowering the sight line; thus the pilot has easy control 
over up and down motions. Similarly, use of the 
rudder will move the sight line to right or left. How¬ 
ever, this will not result in a coordinated turn, but 
will introduce skid. If the pilot sees that the target is 
to the right of the sight line, for instance, the proper 
way of bringing the sight line over — and the only 


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FIRE CONTROL FOR AIRBORNE ROCKETS 


way of bringing the flight direction into agreement 
with the sight line — is to bank, using the rudder to 
keep the ball centered, that is, keep the total force 
in the plane of symmetry. Then the lift vector, in¬ 
stead of being vertically up, will have a component 
to the right, and the airplane will start turning to 
the right, at the proper rate. 

Tracking in azimuth is thus a more complicated 
procedure than tracking in elevation. However, for 
combat pilots the operations needed in controlling the 
plane are almost automatic. Recalling the results of 
the preceding section, we may sum up as follows. 
A pilot can track a ground target fairly well. If the 
sight line is held close to the target for a couple of 
seconds, the flight direction, in elevation, will differ 
from the sight line by not very much more than the 
steady state attack angle. If the airplane has been 
flown very carefully so that no serious skid has en¬ 
tered, the flight direction is close to the sight direc¬ 
tion in azimuth. 

On a calm day, the sight line can be held on a target 
to within 3 to 5 mils. On a rough day, or under bad 
conditions (for instance, in too steep a dive or at too 
great airspeed), the sight line may waver 10 to 20 
mils from the target. Whether the error introduced by 
skid is likely to exceed these figures is not yet known, 
but seems very probable. 

Suppose next that the pilot is tracking with a fixed 
depressed sight. If, for example, there is a computer 
holding the sight line in the plane of symmetry below 
the datum line by the required amount of gravity 
drop for a rocket, the sight feels like a fixed depressed 
sight as far as tracking is concerned. The discussion 
above applies to the present case, with one important 
change: If the pilot wishes to move the sight line to 
the right, for example, and hence banks to the right, 
the immediate result is to carry the sight line in the 
wrong direction, i.e., to the left. For the airplane 
tends to roll approximately about the thrust direc¬ 
tion, and the sight line, being below this direction, 
moves in a direction opposite to the direction of 
banking. The magnitude of this effect has been 
studied. ld The time t B that it takes for the sight line 
to get back to its starting point is indicated by the 
table given below. 5a The assumed conditions are 
v = 169 yd per sec, T = 70 F, dive angle zero and 
5-in. HVAR rocket. 

tB 

r (approximate values) 

1000 yd 0.47 sec 

2000 yd 0.76 sec 

3000 yd 1.07 sec 


There are two things which alter these values 
somewhat. First, the proper lead from the datum or 
thrust line to the sight line differs from \ g by fa, a 
being the attack angle; a is generally negative, which 
would reduce the values of t B above. Next, if a firing 
course is being followed (i.e. a course such that a 
rocket released at any instant will hit the target), 
the airplane’s course is curved downward, which has 
the effect of reducing the lift L; this would increase 
the values of t B somewhat. At any rate, it takes a good 
fraction of a second for the sight line to come back 
to where it started from and get going in the right 
direction, which means that tracking cannot be 
easy in azimuth. 

The above discussion assumed that the turn was 
perfectly coordinated to avoid skid, and that the 
plane was banked suddenly. It should be clear in any 
case that the pilot’s problem of bringing the sight 
line where he wishes it and at the same time of avoid¬ 
ing skid is far from easy. 

When the airplane is banked to the right by the 
amount 0, the lead X in elevation becomes a lead X 
cos 0 in elevation and X sin 0 in azimuth. This helps 
take care of azimuth target motion. 1 ® 

9.4 AIMING PROBLEM 

The lead is the displacement from the datum line 
to that line of sight which is most likely to achieve a 
hit on the target. The lead is compounded of four 
elements: 

1. The displacement from the datum line to the 
effective launcher direction, fa. 

2. The gravity drop \ g . 

3. The parallax correction. 

4. The kinematic lead, that is, the displacement 
which is necessary to compensate for the target mo¬ 
tion relative to the air mass. 

We shall consider now the lead in the plane of sym¬ 
metry of the aircraft. 

9.4.1 Lead in Vertical Plane 

We suppose that the launching aircraft is flying 
without bank; and we consider the lead in the plane 
of symmetry of the aircraft, which is vertical. 

We have already discussed in Section 9.2.1 the 
contribution fa to the lead which is due to the dis¬ 
placement from the datum line to the effective 
launcher line. We have also discussed in Section 9.2.2 
the contribution \ g to the lead due to gravity drop. 


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AIMING PROBLEM 


131 


Strictly speaking the gravity drop \ g should be com¬ 
puted for the range R to the position of the target at 
impact rather than the range r to the position of the 
target at firing (see Section 9.4.3 below). 

The parallax correction is of the order of 

2 2000 . 

— radians =-mils , 

r r 

where r is the range to the target in yards. This is due 
to the fact that the sight is about 2 yd above the 
rocket launcher, so that the sight line must be 
pointed down an extra angle 2/r radians, as is clear 



zontal to the line of sight at the firing instant. Setting 
aside the launcher line, gravity drop, and parallax 
effects, the kinematic lead A* can be obtained by 


A LAUNCHING AIRCRAFT 



Figure 5. Figure for computing kinematic lead. 


applying the law of sines to the triangle ABC of 
Figure 5. Thus 

sin A K _ BC _ v e 
sin a AC V 


Figure 4. Parallax correction. 

from Figure 4. Thus in the case of no wind or target 
motion the formula for total lead A is 

A = A,, + fa H-- 

r 

When wind or target motion is present, the kinematic 
lead A*; must be added to this expression to obtain 
the total lead A. 

9.4.2 Kinematic Lead in Vertical 
Plane 

We shall suppose that the air mass containing the 
trajectory of the rocket moves with a constant ve¬ 
locity. We consider all velocities relative to this air 
mass rather than relative to the earth. The reason for 
this is that the behavior of the rocket is determined 
by the air mass. What the earth is doing relative to 
the air mass or the target or the launching aircraft 
does not enter into the problem. 

We denote the target velocity by v e . (If the target 
happens to be fixed relative to the earth, v e is the 
wind velocity, oppositely directed. In all cases v e is 
to be the target velocity relative to the air mass.) 

We are considering a vertical plane and we assume 
that the target velocity v e is in the horizontal direc¬ 
tion in that plane. Let a be the angle from the hori¬ 


where V is the average speed of the rocket. Since A k 
is small, we replace sin \ K by A k and obtain the ex¬ 
pression 

Ait = — sin a (10) 


for the kinematic lead. Here v e is positive if the target 
is moving away from the launching aircraft. 

The formula for \ K may also be obtained by con¬ 
sidering the target motion across the line of sight; its 
rate is v e sin a. The lead must be such that the rocket’s 
average rate across the line of sight V sin \ K equals 
that of the target. 

Thus the total lead (that is, the angle from the 
datum line to the correct line of sight) is 



(ID 


9.4.3 Approximation for Range R 
to Impact Point 

The range R to the impact point is given approxi¬ 
mately by the formula 

R = r ^1 + ^cos <7^- (12) 

This can be seen from Figure 5 by dropping a per¬ 
pendicular from B onto the side AC or by expressing 
the length of the side r by the law of cosines, apply¬ 
ing the binomial expansion to the resulting expres- 


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FIRE CONTROL FOR AIRBORNE ROCKETS 


sion, neglecting terms of order higher than v e /V 
and replacing cos (a — \k) by cos a. 

9.4.4 Kinematic Lead in More 
General Situations 

A discussion of the kinematic lead in azimuth is 
available. lf It is shown that, at least in certain cases, 
the sight line need only be moved in azimuth through 
80 or 90 per cent of the angle that the flight path 
must be moved through, because of the effect of 
bank. It should be noted that the indeterminacy of 
skid, which affects azimuth errors with any existing 
type of sight, makes it seem undesirable to add 
much complexity to a computer for the sake of that 
dimension. 

Kinematic lead in the plane of symmetry has also 
been studied, taking into account kinematic effects 
on the launcher line term fa and the gravity drop A„. 
It has been shown that under normal circumstances 
the simplified treatment of kinematic lead that we 
have given in Section 9.4.2 is sound. 

9.5 DESIGN OF SIGHTING MECHANISM; 

GENERAL CONSIDERATIONS 

The problem to be considered is the following. 

A computing sight is desired which is simple to 
build, install, and maintain, is as easy for the pilot to 
use as possible, does not restrict his tactics, takes a 
minimum of time to use, and gets the rocket as near 
the target as possible under all conditions. 

It is important to get as full and clear a picture of 
the entire problem as possible. If there is too much 
concentration on one aspect, one is likely to discover 
that a mechanism has been built which will do one part 
of the job beautifully, but fails badly on another part. 
For example, gravity drop being the most obvious 
quantity to be taken account of, and the simplest one 
to study theoretically, there is a great tendency to 
propose sights which correct for gravity drop but do 
nothing about other quantities, such as target mo¬ 
tion. One cannot hope to solve the problem com¬ 
pletely in any simple manner. The more fully one 
tries to achieve the aims of accuracy and tactical 
freedom, the more complicated the apparatus is 
likely to become, not to mention the probable in¬ 
crease in weight and space taken. One should prob¬ 
ably strive for some kind of compromise, sacrificing 
some points that seem of less importance to keep the 
mechanism within reasonable bounds. 

The essential difficulty of the problem is apparent 


from the following consideration. A much used form 
of attack is to fly in toward the target at tree top 
level, remaining hidden as long as possible; at the 
last instant, one rises up, puts the sight on the target, 
fires, and pulls away. One cannot hope to get an 
accurate determination of range in such an attack; 
hence, unless one fires from quite close range (which, 
as a matter of fact, is usually done), one cannot ex¬ 
pect an accurate value of the gravity drop. 

Another fundamental difficulty is that caused by 
the tendency of an airplane to skid. It seems fairly 
certain that skidding of the order of 20 mils is hard 
to avoid. The seriousness of the resulting error has 
been discussed in Sections 9.2.1 and 9.3.2. If one is 
doing lead computing with a gyro sight, there is 
likely to be some correction for skid; perhaps a third 
of the error will be removed. 

9.5.1 Possibility of Simple 
Computers 

It has been common practice either to aim rockets 
by eye, or to strive to attain a standard set of con¬ 
ditions (of dive angle, airspeed and range), and fire, 
using a fixed lead set in beforehand. It must be real¬ 
ized that with long training and practice, pilots can 
become fairly accurate in their firing. A good tennis 
player can return a ball fairly closely to where he 
wants it; it would be quite difficult to construct a 
computer that would move to the right position and 
hit the ball in the right way to make it do the same 
thing. 

If a simple computer is designed to do part of the 
job and let the pilot do the rest, it must be designed 
so as not to upset the pilot’s normal reactions to the 
situation; for example, it must not replace a train of 
habitual thought by a set of mental numerical calcu¬ 
lations. 

Suppose first no computer is used. Pilots can be 
trained to dive at a given dive angle without very 
large errors; it is more difficult for a pilot to decide 
when he is at a given range. Probably distances at the 
target can be estimated much better than the range 
to the target. 

The simplest kind of a computer that would be a 
real help would probably use altitude above the tar¬ 
get (from altimeter and hand set target altitude), 
dive angle (from an accelerometer, to be used while 
a fixed sight is held on the target 18 ), and indicated 
airspeed to compute gravity drop and attack angle, 
and feed this in some manner into the sight line. 


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DESIGN OF SIGHTING MECHANISM; GENERAL CONSIDERATIONS 


133 


If there is a gyro lead computing sight in the air¬ 
craft (for guns), it may be used with a large constant 
time-of-flight setting to give a good average value of 
gravity drop and attack angle (for a not too large 
range of dive angle), and to take some account of 
wind and target motion (see Section 9.6.4 below). 

9.5.2 Comparative Importance of 
Different Components of Lead 

We now consider what must be done to obtain a 
fairly complete solution of the rocket sighting prob¬ 
lem. We shall study the problem mainly in the 
vertical plane, since the principal difficulties (except 
for skid) are encountered there. 

The formula for the total required lead is equation 
(11). For present purposes, we may omit the small 
parallax term 2/r. Thus we consider: 

\ = \ g +fa+ V -^^ • (13) 

The angle X is the angle from the datum line of the 
plane measured downward to the sight line. Thus X 
is positive when the sight line is below the datum 
line. (Writing the formula this way does not mean 
that the datum line has to be used as the reference 
direction. In terms of another direction, constant or 
variable relative to the aircraft, the required formula 
may be deduced from equation (13).) 

A rough estimate of the sizes of the three terms is 
important in determining how careful one must be in 
taking care of each of them. 

The gravity drop term \ g is roughly 10 mils, plus 
15 to 20 mils for each thousand yards of horizontal 
distance to the target. (See Section 9.2.2 and Figure 
6 below.) Thus if one wishes to fire from any hori¬ 
zontal distance ranging from 3,000 yd in (with the 
5-in. HVAR), \ g will vary roughly from 65 to 20 mils, 
a variation of 45 mils. 

The attack angle term fa depends primarily on the 
airspeed and dive angle. Suppose the likely ranges of 
airspeed are from 240 knots = 135 yd per sec to 
380 knots = 214 yd per sec, and of dive angle, from 
20° to 60°. Taking the F6F aircraft, with b = 71.3 yd 
and a = —0.043 radians, and the 5-in. HVAR, pro¬ 
pellant temperature 70 F, we find from equation (5) 
(setting if, = 0), and equation (1) (or from the 
“tables” 5 ) 

a = —3.6 mils, / = 0.82, at 240 knots, 20° dive, 

a — —34.6 mils, / = 0.94, at 380 knots, 60° dive. 


Thus fa varies about from —3 to —34 mils, a 31 mil 
variation. 

The kinematic lead v e sin a/V may be very large for 
fast targets on windy days, and negligible for fixed 
targets on very calm days. Let us take a 30-mph tar¬ 
get motion relative to the earth, with an equal wind 
in the opposite direction, as a typical value of v e ; then 
v e = 60 mph = 29.3 yd per sec. If <r = 60°, v e sin 
a = 25.4 yd per sec. Taking V = 500 yd per sec (see 
Section 9.2.3) gives 


Since the kinematic lead may be in any direction, we 
have a possible variation of 100 mils in its value. 
Thus on windy days, with moving targets, the kinematic 
lead can be the most important item to be taken care of; 
on calm days, with targets at rest, it is relatively unim¬ 
portant. 

9.5.3 Graphs of Lead 

To help the reader’s feeling for the dependence of 
lead on range and dive angle, we give some graphs of 
lead as required on a firing course, that is, a course 
at each instant of which the rocket may effectively 
be launched. In each graph, contour lines of constant 
lead (in mils) are plotted on a polar diagram in which 
the radius vector is the range r and the polar angle is 
the sight angle a. Thus the graph coordinates are 
similar to actual conditions. All graphs are for the 
5-in. HVAR, propellant temperature 70 F, and for 
plane characteristics a 0 = —0.043, b = 71.3, as for 
the F6F aircraft. 

Figure 6 shows the dependence of \ g + 2/r on 
range and dive angle, other quantities being fixed. 
It shows how, to a reasonable approximation, the re¬ 
quired lead is a function of horizontal range x = r 
cos a, if x is not small. At shortest ranges, the paral¬ 
lax correction 2/r has a considerable effect in curving 
the contour lines. 

If the computer is to compute gravity drop, paral¬ 
lax, and launcher line term, at a given airspeed, it 
must produce the leads indicated in Figure 7. The 
fact that angle of attack a depends on dive angle has 
the effect of tipping the contours considerably. 

Figure 8 would not be used in a computer contain¬ 
ing an airspeed input. The most likely airspeeds dur¬ 
ing a dive will be higher for steeper dive angles; this 
will decrease the lead required in the steeper dives. 
Typical indicated airspeeds are 300 knots for a dive 


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134 


FIRE CONTROL FOR AIRBORNE ROCKETS 



Figure 6. Gravity drop and parallax (fixed airspeed). Firing course. Contours of \g + 2 ,/r in mils. 5.0" HVAR. v — 300 
knots, T = 70 F. 


angle 8 = 30°, and 375 knots for 5 = 00°. In the 
graph, we have taken 

Vi = 225 + 2.55, 

where 5 is measured in degrees and Vi in knots. Ac¬ 
cordingly the graph shows appropriate values of X 
at each range and dive angle. 

9.5.4 Computing Kinematic Lead 

The pilot’s ideal would be to put the pip on the 
target, then fire at once, or wait as long as he likes 
before firing. In other words, he would like: (a) an 
instantaneous solution of the problem, so that the 
moment the sight line is on the target, the correct 


lead is obtained, and (b) a continuous solution, so 
that the lead continues to be correct as long as he 
holds the sight line on the target — he is following a 
firing course. 

The first thing to be noted is that an instantaneous 
solution of the problem, including kinematic lead, is 
impossible, if information from outside sources about 
wind and target motion is not used. This fact is 
simply demonstrated by the following example. 
Suppose the pilot flies out of a cloud, and finds he is 
aiming at a tank on a bridge. He fires at once, hoping 
to destroy both tank and bridge. By the time the 
projectile arrives on the ground, the tank and bridge 
are some distance apart; the rocket cannot be ex¬ 
pected to blow them both up. 


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DESIGN OF SIGHTING MECHANISM; GENERAL CONSIDERATIONS 


135 



RANGE IN YAROS 


Figure 7. Total lead (no wind or target motion), fixed indicated airspeed. Firing course. Contours of X in mils. X = 
\g + fot + 2/r. 5.0" HVAR, F6F. v { = 300 knots, T = 70 F. 


The mere fact that kinematic lead is due to chang¬ 
ing conditions — rate of change of position of the 
target relative to the air mass surrounding the plane 
— shows that it can be determined only by noting 
the change over a period of time. If it is assumed that 
the pilot’s only contact with the target is visual, and 
use of this information is made by holding a sight 
line on the target, it is seen that the sight must be on 
the target during a definite period of time if kinematic 
lead is to be determined. 

Since a sight line cannot be held perfectly on the 
target, the rate of motion of the sight line at a given 
moment is apt to be quite different from the angular 
rate of the target direction at that moment. Hence 
the sight line will have to be held on the target long 


enough so that some kind of average sight rate will 
be near the actual target rate. 

Let us say the kinematic lead is to be determined 
within 5 mils, and tracking errors of the order of 3 
mils may be expected. We wish to find out what 
lower limit this puts on the tracking time. Suppose, 
for instance, that the target rate is determined by 
comparing the target direction at two instants ti and 
t 2 . If the sight line is, say, 3 mils off in one direction 
at the start and 2 mils off in the other direction at the 
end, there is an error of 5 mils over the time interval 
At = t 2 — h } making an error of 

0.005 

Aak = ^ ~ radians per second 


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136 


FIRE CONTROL FOR AIRBORNE ROCKETS 



Figure 8. Total lead, variable airspeed (no wind or target motion). Firing course — variable velocity. Contours of 
X in mils. X = \g +/« + 2/r. 5.0" HVAR, F6F. v t variable, T = 70 F. 


in the rate a k of the sight line, which is attributed to 
kinematic lead. Now the sight rate <r k due to wind and 
target motion, and the kinematic lead A*, are given 
by 

v e sin a v e sin a r . 

a k = - > A k = — = ~ a k) 

r v V 

hence the error Act*, in a k causes an error A\ k in kine¬ 
matic lead given by 

r a r 0.005 r 0.01 
AX, = y Aa k = — M = 1(J00 — ■ 

(This assumes a fixed sight is being held on the tar¬ 
get. The figures would not be very different if the 
sight line were being moved to take account of kine¬ 


matic lead.) Hence, if we wish A\ k < 0.005, we must 
have 

0.01 r ^ r 
At = ~A\ k 1000 > 2 1000 ' 

Thus at 2,000 yd range, we must take at least 4 sec 
for tracking. Note that this is entirely independent of 
what sort of computer is used. 

The above was based on the assumption that the 
computer started functioning at the instant t\. Since 
the computer cannot know at what instant the sight 
line has settled on the target, this requires an oper¬ 
ation on the part of the pilot (say the pressing of a 
button) to start the computation of kinematic lead. 
Of course, there might be some automatic way of 


CONFIDENTIAL 














DESIGN OF SIGHTING MECHANISM; GENERAL CONSIDERATIONS 


137 


starting the computation at approximately the right 
moment; for example, it might start functioning 
as soon as the angular rate has gotten below a certain 
value. 

Probably a lagged value of the angular rate lh will 
be used rather than the angular change over the 
time taken. Then momentary oscillations of the sight 
line may have a less harmful effect; but also there 
may be difficulty with the transient term. If the 
computing is started too soon, the angular rate due 
to the sight line getting onto the target will give a 
false picture of kinematic lead unless tracking is 
continued long enough for this to die out. In any 
case, it is unlikely that the kinematic lead can be 
calculated better than to about 5 or more mils in 
4 sec if the range is over 2,000 yd. 

9.5.5 Sighting Procedures 

We study here the relation between what the com¬ 
puter needs in order to compute the lead and what 
the pilot must do to enable the computer to work 
correctly. We saw in the last section that the sight 
must be held on the target for some seconds if the 
kinematic lead is to be computed. To obtain the 
gravity drop within 5 mils, one must know what is 
essentially horizontal range (see Figure 6) to within 
about 300 yd. The computed value of this variable 
may or may not be instantaneous. The computation 
of attack angle will probably be done as quickly as 
that of gravity drop. 

Thus it is likely that the pilot’s procedure will in¬ 
clude holding a sight line on the target over a period 
of time, say several seconds. We now raise the 
question: What is the relation of this period of time 
to the moment of firing the rocket? If a firing course 
is to be followed, as in (b) of the last section, then 
the moment of firing lies anywhere during the track¬ 
ing time, after the first few seconds. This suggests 
that the computation of lead should be taking place 
at the same time that this lead is being employed to 
bring the plane into the correct position. Hence, if 
the motion of the plane is itself being employed in 
the computation of lead, there is a circular process: 
computation of lead affects motion of the plane which 
affects computation of lead. This may lead to track¬ 
ing instability, or at least a slowing down of the 
computation (see the next section). 

In the design of a computer, such slowing down 
must be avoided as far as possible. The problem may 
be avoided by not letting the computation process 


affect the sight line, for example, by computing the 
proper lead while a fixed sight is held on the target. 
In this case either (a) one> must arrive at the correct 
firing position at some moment during the process, at 
which moment the pilot must be told to fire or firing 
is automatic, or else (b) the pilot must bring the 
plane into firing position after the computation of 
lead is accomplished. 

We give some examples of how (a) or (b) above 
may be accomplished. Suppose, at the start of track¬ 
ing, the fixed sight has a lead which is somewhat less 
than that required for firing at the moment. As the 
pilot approaches the target, the correct lead di¬ 
minishes; the computer is working out this lead; 
at the moment it decides that the desired lead 
equals that which the sight has, it fires the round 
automatically. 

There are clearly some disadvantages in this pro¬ 
cedure. The starting conditions must be arranged in 
some fashion so that the fixed lead will come about 
where the pilot expects; either he must be able to 
arrange to have the firing position (and hence the 
amount of the fixed lead) about where he desires it, 
or he must be able to know he is starting beyond his 
firing position. Having once started, he may not 
know how long he will have to wait, and he may 
begin to fear that he started too close, unless the 
computer tells him otherwise in some fashion. 

These difficulties might be somewhat alleviated by 
letting the lead be variable, for example, letting it 
start at zero or a negative quantity, and gradually 
increase. Then the pilot will be sure that the proper 
lead will be reached before long. Moreover, it will 
take longer, and thus the computer will have more 
time to function properly, at the longer ranges 
(where more lead is required), which is desirable. 
(See Section 9.6.4.) 

In case (b), one Avould like the procedure of bring¬ 
ing the plane into position and firing to be automatic. 
This can be accomplished by tossing , 18,23 which has 
the added advantage of combining the maneuver 
with the process of getting away. The procedure in 
brief is as follows. When the pilot has tracked for 
some seconds, he presses a button, which stops the 
computation of lead and starts the toss mechanism 
functioning. The pilot now pulls the nose of the plane 
up. The angle A<f>, through which the flight path must 
turn, having been determined, the instrument now 
measures the normal acceleration (or perhaps g cos 
8 —integrates this to obtain vA(f>, and releases the 
rocket when A <j> has reached the desired value. 21-23,25-27 


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138 


FIRE CONTROL FOR AIRBORNE ROCKETS 


9 . 5.6 Operating Stability 

The above discussion should be sufficient to show 
that the manner of computing lead is tied up with the 
pilot's operating procedure, and the theory of the com¬ 
puter must go along with the operating procedure. 

Thus when the design of the mechanism has been 
determined, it is necessary to make a careful analysis 
of the way the mechanism will perform under actual 
conditions. In particular, any unevenness in the in¬ 
puts, due to a variety of causes such as bumpy air, 
must be considered. 

In this section, we shall study only the likely effect 
of imperfect tracking on the computation of lead 
and the resulting tendency to instability. Thus we 
consider here only effects resulting from false angular 
rates; if the mechanism operates independently of 
angular rates, there will be no tracking instability. 
(Another example of likely instability is that due to 
the use of an accelerometer. 18 If the instrument is 
very sensitive to accelerations and acts on the sight 
line gently, the accelerations due to bumpy air 
might cause the sight line to waver so that it cannot 
be held on the target.) 

Consider the computation of kinematic lead, pre¬ 
sumably based on the angular rate of the plane, or at 
least of the sight line. To see the effect of the false 
angular rate produced by getting on the target, let 
us take a numerical example. Suppose, in order to 
watch the target area, the pilot starts by diving 5° too 
steeply. Let us say he takes 1.5 sec to pull up the 5° 
to get on target. The average angular rate is then 
(5°/57.3°)/1.5 radians per sec = 58 mils per sec. 
If the range is 2,500 yd, this gives a hypothetical 
target velocity 

v e sin <t = ra k = 145 yd per sec 

(see Section 9.5.4), for which the proper kinematic 
lead would be in the neighborhood of 


which is far beyond reality. If this lead were actually 
fed into the sight line, the pilot would have to pull up 
290 mils = 16.5°. If he did so, and stopped there, 
the computer would now decide that no lead was 
required, and the pilot would start pulling the nose 
down again, putting in a false angular rate in the 
opposite direction; clearly the situation is unstable. 

If the pilot were nearly on the target, and started 
pulling up the little bit necessary to get on, and if 
the computer worked rapidly, it would at once decide 


that considerable lead was required and make him 
pull up a lot more. The same instability is present. 

The usual way around this difficulty is to have 
the computer use a lagged value of the angular rate 
to compute kinematic lead; see Section 9.6.3. By 
proper choice of the constants, the process is rendered 
perfectly stable; the primary remaining difficulty is 
the slowness of the computation. A limit on the speed 
of computation was derived in Section 9.5.4; making 
the tracking process stable may slow the computa¬ 
tion further. 

The above discussion illustrates some possible 
causes of instability; the whole problem has to be 
studied for any given computer. 

9 . 5.7 General Summary: The 
Search for a Satisfactory 
Computer 

The designer of an aircraft rocket sight is con¬ 
fronted with enormous difficulties. In brief: 

1. The corrections that are to be made for the 
various quantities are difficult to determine, par¬ 
ticularly for a fighter plane with no operator other 
than the pilot. 

2. Assuming certain corrections are to be made, 
it is very hard for the pilot or a mechanism to know 
whether they are being made properly, primarily 
since the direction in which a fin stabilized rocket 
starts out is approximately the flight direction rela¬ 
tive to the surrounding air mass, an elusive variable. 

3. Solving problems of such difficulty is likely to 
involve complicated mechanisms and complicated 
procedures; but it is essential that at least the oper¬ 
ating procedure be kept as simple as possible. 

How should one attack the problem of designing 
a sight? One may attempt something fairly standard 
in principle, or one may try something which is 
essentially new. An inventor who brings to the 
problem a deep insight or a really novel approach 
cannot be greatly helped by advice. If he is a pilot, 
he may best make himself thoroughly familiar with 
the problem by firing rockets himself, before he 
formulates his ideas. But when ideas for a solution 
are once formulated, then a study of the most im¬ 
portant considerations is in order. It is at this and 
later stages of the problem that the work of AMP 
may prove of future service. It should help find 
omissions in the theory, help relate the theory to 
known theories, and help remind the designer what 
things to watch for. 


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THE DESIGN OF A SIGHTING MECHANISM 


139 


The fundamental question that must be kept in 
mind about a gadget is, how will it work when actually 
put to usef This must be studied thoroughly back in 
the design stage, particularly in the present problem, 
where a pilot, with his life in danger, has to think of 
a myriad of things in a few seconds if he is to accom¬ 
plish his mission and come back alive. 

9.6 THE DESIGN OF A SIGHTING 
MECHANISM 

The quantities that appear in the lead formula (13) 
are determined by the values of certain other quanti¬ 
ties, as is implied by the formulas that we have 
written. The following table gives an indication of 
this dependence. 


We have not listed angle of attack a as an input. 
While it is true that it may become possible to meas¬ 
ure the aircraft’s angle of alttack directly (by differ¬ 
ential pressure methods, for example), such a 
measurement wohld appear to offer no advantage 
over a computation of the rocket’s angle of attack 
from formula (5), with <j> taken as zero. Observe that 
it is the rocket’s angle of attack that is pertinent. 
Properly calibrating a measuring device appears as 
difficult as determining the correct constants for 
formula (5). 

A discussion has been given 11 of the potentialities 
of certain devices for measuring altitude, indicated 
airspeed, time intervals, radar range, attitude of 
launching aircraft (from a gyro), angular rates (from 
gyros), angles of attack and skid, linear accelerations, 


Function 
\ g , gravity drop. 


R 

f, launching factor. 

Vi 

v 

a, angle of attack. 

v e sin <r, target velocity across line of 
sight in the vertical plane. 

V average rocket speed. 


Variables on which it 
depends critically 

R , range from launching point to posi¬ 
tion of target at impact. 

r, range from launching point to posi¬ 
tion of target at firing instant. 

v 

Vi 

Vi 

8, angle of depression of datum line. 

v 


Variables on which it 
depends less critically 

7 , angle of depression of effective 
launcher line. 
v, own speed. 

T, propellant temperature. 

<r, angle of depression of sight line. 
v e /V, ratio of enemy speed to average 
rocket speed. 

V{, indicated own speed T 
h, altitude. 
h 
v 

4>, rate of turn of flight path in vertical 
plane. 


This table refers only to one aspect of the aiming 
problem: the component of lead in the plane of sym¬ 
metry of the launching aircraft, assuming no skid or 
bank. (If bank (3 were present, it would be necessary 
to replace g by g cos in the formula for a and to 
replace \ g by \ g cos /3.) The lead in the wing plane of 
the aircraft should be considered; it depends criti¬ 
cally on the skid and bank. 

We see from the table that a computer could pro¬ 
duce the lead X in the plane of symmetry if it had 
estimates of 

r, v e sin a, 8, Vi ; 

T , h or v, o', 4 >• 

(In determining \ g , the angle y may be replaced by 
8.) The fact that these variables are sufficient does 
not imply that a machine must use them explicitly. 


angular accelerations, weight of aircraft, and pro¬ 
pellant temperature. There is a discussion also, for 
example, of the determination of range from altitude 
difference and dive angle ( 8 , y, </>, or a), dive angle 
from normal acceleration, the correction of an accel¬ 
erometer for attack angle, dive angle from a forward 
accelerometer, dive angle from the rate of decrease 
of altitude, range from angular rates, attack angle 
from airspeed and dive angle. 

9.6.1 Estimates of Situation by 
Pilot — Firing Projectiles for 
Information 

The pilot, while flying, may make various esti¬ 
mates of variables affecting the sighting problem; he 


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140 


FIRE CONTROL FOR AIRBORNE ROCKETS 


may put his estimates into a computer, or he may try 
to fly so that these variables take on prescribed 
values. Examples of such variables are: dive angle, 
range to the target, expected airspeed, magnitude 
and direction of wind and target motion. Contriv¬ 
ances may be used to help him in his estimates. For 
example, if the target is of known dimensions, he 
may span the target with a reticle of variable size; 
the reticle size then determines the range. If the pilot 
flies level and at right angles to the direction to the 
target, and tips his wing over so that it lies just below 
the target, he may take his angle of bank as the 
dive angle to the target (if he goes at once into the 
dive), and read this angle from the gyro horizon. 
Or again, if he flies level without banking until the 
target is visible just ahead of some point of the wing, 
this point determines the dive angle to the target. 

Firing cannon or machine guns at the target, or 
along a line running up to the target, furnishes the 
pilot with information which can be helpful in aim¬ 
ing rockets. Or again, watching where the first 
rockets land, the pilot may attempt to correct his aim 
with later rockets. In this regard, it should be noted 
that the plane will be nearly half way to the target 
before rockets with a corrected aim may be fired, 
and the conditions (range, wind) may then be con¬ 
siderably different from what they were before; 
moreover, the pilot will not know what the cause of 
the miss was, and the required correction depends on 
the cause of the miss. 

By a proper approach the pilot may be able to 
make the azimuth lead small or zero. 

If the target is fixed relative to the earth and if the 
wind velocity is known, the target velocity is then 
known. 

9.6.2 Rate of Rotation a of Sight Line 

We continue to study the problem in a vertical 
plane. The angular rate of rotation 

do 

co = o- = — 
dt 

of the sight line is very likely to be one of the quanti¬ 
ties involved in future sights. It is clear from Figure 9 
that 

v sin (c — <pi) — v e sin o 

co = a = - > 

r 

the first term being the contribution of own speed, 
the second, that of target motion. Now 

cr — 4> = \ — a, 


AT LAUNCHING AIRCRAFT 



Figure 9. Figure for computing lead in terms of 
angular rate of rotation of sight line. 


(see Figure 9). This is a small angle; we replace its 
sine by its radian measure. Thus 

r'o = v(\ — a) — v e sin o. (14) 

This relation may be used with equation (13) to 
eliminate v e sin cr, r or a. Eliminating v e sin a gives: 

V r 


, fV ~ V 4 - 
A = -- Oi + 


V - v 


We put 


h = 


V - v X ° V-v a * 


fV -v 


V - v 

This quantity is near unity, since 

fV — v 1 - / 


l-h=l- 

V — v 

which is near zero. We write 


1 - 


(15) 

( 16 ) 


V R = V — v ; (17) 

thus Vr is the average speed of the rocket relative to 
its launching aircraft. 

Thus equation (15) becomes 

X = ha + —-X fl — — <7 . (18) 

Vr Vr 


The coefficient h is given by equation (16) and is 
close to unity. 


9.6.3 Computing Sight which 
Predicts Full Lead in 
Vertical Plane 

The mechanism to be described is of the same sort 
as the Army Draper-Davis sight and the Navy 
PUSS, 4 when either is used for rockets. 

We shall suppose that range r will be measured by 
a radar mechanism and will be available for the com¬ 
putation. With estimates or approximate measure¬ 
ments of dive angle, propellant temperature, and 


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THE DESIGN OF A SIGHTING MECHANISM 


141 


own speed, the knowledge of range r permits a satis¬ 
factory mechanization to produce the gravity drop \ g . 

The angle of attack a can be computed mechani¬ 
cally from a measurement or estimate of indicated 
own speed and dive angle. The speeds V and Vr are 
determined by own speed v in a linear relationship. 
Thus, in principle, all the quantities in equation (18), 
except <x, can be considered accessible. However, a 
mechanism for computing all the quantities that we 
have here discussed according to the equations given 
would be overcomplicated. In practice, average val¬ 
ues would be used instead of certain variables, and 
engineering compromises would be appropriate. 24 



To measure o-, a gyro could be mounted with axis 
(always in the plane of symmetry of the aircraft) at 
an angle rj below the horizontal. As in the usual rate 
sight, the gyro, sight line, and datum line are coupled 
so that 

7] = a + cX = (1 + c)a — c8, (19) 


pated average value of X. If one puts X = 0 in the 
expression for /3, the gyro equation (20) reduces to 

V r 

cuX + X = hex + — \ g — — <j , (22) 

V r V R 

where 


is the sensitivity of the gyro system. 

The differential equation (22) implies that the 
mechanization has the following properties: 

1. X at each instant is such as to make X change in 
the direction which brings X closer to the right hand 
member (which is the true lead), the rate of change 
increasing with the discrepancy. This can be seen by 
solving equation (22) for X and observing that cu is 
positive. Thus the solution will be a lagged value of 
the correct lead (equation (23)). 

2. The transient is eliminated after about 2 cu sec. 
In the Draper-Davis sight, c = 0.2; in PUSS, 
c = 0.25. 

3. If the sight line is subject to disturbances, the 
resulting disturbance in X will not be unduly large 
compared to the disturbance in <r even though the 
disturbance in cr be large (as it may, indeed, be) . 30 

The optimum choice of the functions u and 0 has 
been discussed 2b as well as the errors which can be 
expected from such choices of u and jS, under ideal 
conditions. 

9.6.4 Sights Whose Prediction 
Neglects Kinematic Lead in 
Whole or in Part 


where c is a positive constant (often called — a ). The 
gyro is made to precess toward an equilibrium posi¬ 
tion which is at an angle (3 below the datum line, at a 
rate proportional to the angular displacement: 

fj = k(8 + jS — rj) = k(5 + (3 — a — cX) 

= - (c + 1)A), (20) 

where A; is a positive constant. Now direct substitution 
shows that equation (20) reduces to equation (18), 
if k and /3 have the following values: 

Vr r V r V 1 

k = i & = (c + 1) —- \ g + ha + c — X • 

(c + 1 )r [Vr V r J 

( 21 ) 


In this section we suppose that the target motion 
relative to the air mass is neglected. In the case of 
targets fixed relative to the earth, this implies that 
wind is neglected. 

When v e vanishes the equation (14) for sight rate <r 
becomes 

a = — (X — a) . (24) 

r 

The equation (13) becomes 

A =5 fa + A g . (25) 

If the launching factor / were unity, these relations 
would imply that A — a = \ Q and 


All the quantities used in defining k and (3 may be 
considered known, except X. One may use an antici- 



(26) 


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142 


FIRE CONTROL FOR AIRBORNE ROCKETS 


Now equation (2) implies that 

\ g — (A 0 + Air) cos (f>, 

since/ = 1 implies that <f> = 7. If we take for example 
A q = 0.013, Ai = 0.000016, and user = r/1000, this 
gives 

. / 0.013X 

<j = (0.016 + —~ J v cos <j> mils per sec, (27) 

where v is in yards per second. If v = 169 yd per 
sec = 300 knots, <f> = 40°, r = 2,000 yd, then <7 = 
2.9 mils per sec. At medium or long ranges , and at a 
given dive angle , this shows that <7 is nearly constant. 

This fact may be used in the following way. Use a 
gyro to turn a sight line at the rate shown above. 
Then if the sight line is held on the target, <7 will have 
the above value, and hence the lead has the correct 
value. Note that (assuming / = 1) the attack angle 
need not be known, since it does not occur in the 
formula. Thus this method uses “target coordinates,” 
and is independent of what the air mass may be 
doing. 

This method is useful if bombs are being dropped, 
in which case / —- 1 and A 0 = 0; but in the case of 
rockets, the method results in a large overcorrection 
of kinematic lead, since <7 will be appreciably incor¬ 
rect when target motion is actually present. This 
overcorrection could be taken care of as follows. 
First, fly a pursuit course, holding a fixed sight on the 
target; measure the angular rate; then fly the course 
described above, but making a correction dependent 
on the angular rate that was found. 

As a modification of the above method, one could 
use a coupling of the gyro to axes in the plane; instead 
of the equation (27) 

<7 — CO, 

use 

<j = co -J- k'd. (28) 

This may be mechanized by a slight alteration of the 
usual rate sight. The same essential features are 
present as in the former case, but tracking will be 
easier. 

One may make a sort of compromise between the 
above method and that of Section 9.6.3 by using the 
rate sight of Section 9.6.3 with a fixed sensitivity 
(to take care of kinematic lead at an average range), 
and with a fixed position of equilibrium of the gyro 
(to give an average gravity drop). This actually is 
much more successful than would appear at first 


sight, for the following reason. At long ranges, the 
lead is too small; at some intermediate range, the 
correct lead will have decreased to the lead actually 
used. As the range decreases further, the angular 
rate due to the velocity of the plane, given by equa¬ 
tion (26) if v e = 0, / = 1, increases since r is de¬ 
creasing. The gyro lags behind, making the sight lag 
also (relative to the plane), thus decreasing the lead 
actually obtained. Thus the correct lead and the lead 
obtained are both decreasing. The actual curve of 
lead obtained may be made to fit the curve of desired 
gravity drop without very large errors, say if the 
range is less than 2,500 yd. At short ranges, the 
mechanism is apt to overcompute greatly. The dive 
angle must be taken care of separately. 

The British method of using the Mark 23 sight is 
precisely of the sort just described. It has been 
studied in detail. 2a Optimum settings and errors under 
ideal conditions are discussed. 


9.6.5 Peanut as a Rocket Sight 


The sights considered thus far have been designed 
for firing courses, courses during which the rocket 
could be released at any instant, after the computer 
had settled down. The sight Peanut is of a different 
type: It automatically releases the rocket when a 
fixed lead, previously set into the sight, is the correct 
lead, in the absence of target motion. This sight has 
been discussed in detail. 20 Peanut is based on the 
equation 


B\v{\ — a) 
\ — fa — B 0 


(29) 


obtained by writing 


Xa = Bq + Bir 


and eliminating \ g and r from equations (24) and (25). 
The quantities B 0 and Bi are proportional to cos 7. 
[See equation (2).] A fixed X is put into the mecha¬ 
nism, which then releases the rocket when a smoothed 
value of b assumes the value given by equation (29). 

In the case of bombs, / = 1 and B 0 = 0, so that 
equation (29) reduces to 


b — B\V. (30) 

A sight like Peanut is open to the objection that 
the pilot cannot fire when he feels it appropriate to 
fire. He must wait for the release and he may be un¬ 
able to judge when the release will come. In a strong 
head wind the release may be delayed dangerously. 


CONFIDENTIAL 




PART III 


ANTIAIRCRAFT ANALYSIS 


CONFIDENTIAL 




Chapter 10 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


10.1 INTRODUCTORY REMARKS 

i vision 7 of the NDRC, in charge of research 
and development in the whole field of fire con¬ 
trol, was fortunate in having, both on its own 
technical staff and as employees of its contractors, a 
considerable number of individuals who were highly 
expert and experienced in the analytical aspects of 
fire control. Thus it is natural and proper that the 
well-balanced and broad program on antiaircraft 
equipment, including the more mathematical aspects, 
was carried out by Division 7, and will be found re¬ 
ported in their technical papers. 

The Applied Mathematics Panel was, nevertheless, 
asked from time to time to do various jobs of analysis 
referring to antiaircraft equipment. The present 
chapter reports the AMP activities in this field. And 
since the AMP activities were of a rather assorted 
and catch-as-catch-can nature, it has seemed de¬ 
sirable to preface the chapter with some brief general 
comments on antiaircraft problems and equipment, 
in the hope that these general remarks will furnish a 
background against which the separate — and neces¬ 
sarily somewhat disconnected — sections of this 
chapter will be more understandable. 

As far as the words themselves are concerned, the 
phrase “antiaircraft fire” should include all fire di¬ 
rected against aircraft. Military usage, however, 
seems to restrict this phrase to fire directed against 
aircraft from guns located on ground or ship, the 
phrase “plane-to-plane fire” being more common for 
fire directed from one aircraft against another. 

The present chapter follows accepted usage, and 
thus refers to ground-to-plane or ship-to-plane anti¬ 
aircraft fire. The phrase “antiaircraft fire control” is 
often interpreted as including various weapons, such 
as rockets and guided missiles, in addition to the 
orthodox guns of various calibers. The present chap¬ 
ter, however, refers only to antiaircraft guns. It is 
customary to divide antiaircraft guns into two main 
classes. Automatic weapons include machine guns of 


various calibers which utilize belted ammunition, 
and certain larger guns such as the 20-mm, and the 
40-mm Bofors which automatically fire clips of shells. 
Automatic weapons normally fire solid slugs, or pro¬ 
jectiles with contact fuzes, but they do not employ 
projectiles with time fuzes, nor does it seem likely 
that their dimensions would ever permit the use of 
proximity fuzes. Major caliber antiaircraft guns 
(sometimes called heavy antiaircraft) include the 
various larger sized guns which load single shells of 
calibers 3 in., 5 in., 90 mm, etc., these projectiles 
normally being time-fuzed. 

An antiaircraft fire control system involves the 
following. 

1. Target data gathering. 

2. Transmission of these data to predicting mecha¬ 
nism.* 1 

3. Prediction; or estimation or calculation of gun 
orders (i.e. position of gun and fuze setting if this is 
involved). 

4. Transmission of these gun orders to gun. 

5. Orientation of gun (and setting of fuze, if in¬ 
volved) in accordance with these gun orders, and 
firing of gun. 

In the case of a strafed infantry man, firing at a 
plane with his rifle, the target data are collected by 
his eyes, transmitted by his nervous system to his 
brain, which estimates the proper position for his 
gun. This estimate is again transmitted by the nerv¬ 
ous system to muscles which orient the gun and press 
the trigger. The elements of a complete fire control 


a In the older terminology of the Army, the predicting 
mechanism, usually containing the mechanisms for obtaining 
angular target data but not target range, was known as a 
director. In the Navy, the word director usually refers to the 
target data mechanism, and the predictor is called a computer. 
In more recent Army equipment, the target data device is 
called a tracker and the predictor a computer (as in the Navy). 
To avoid this confusion of terms, we will use the general, and 
hardly misinterpretable, term predictor for the device which 
accepts target data and predicts the future position of the 
target. 



CONFIDENTIAL 


145 



146 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


system are all present, but the system is unmecha¬ 
nized and wholly personal in character. 

In the intermediate case of a 40-mm gun, the target 
data are obtained by two men who, one working in 
elevation and one in azimuth, ‘ ‘track’ ’ the target by 
keeping the cross hairs of telescopes aligned on it. 
These tracking telescopes are an integral part of the 
predicting mechanism, so that transmission of target 
data presents no problem. A mechanical (and electri¬ 
cal) device accepts these target data, and continu¬ 
ously computes what position the gun should have 
in order that projectiles fired from it meet the target 
in space. These gun orders are transmitted by an 
electrical data transmission (selsyn) system to the 
hydraulic or electric servos which, obeying the signals 
they receive, automatically keep the gun properly 
oriented. The tracking and predicting mechanism is 
normally some distance from the gun, to avoid shock, 
smoke, etc., so that transmission of gun orders from 
prediction to gun constitutes a real (although satis¬ 
factorily solved) problem. 

In a still more completely mechanized system, the 
target data might be obtained from a radar set which 
follows the target automatically, once its beam is 
placed on the target, and which makes continuously 
available the angular azimuth and elevation of the 
target and its slant range in yards. These data are 
then transmitted by a selsyn or similar system to the 
predicting mechanism which computes the gun 
orders. The gun orders are transmitted electrically to 
the servos, hydraulic or electric, which position the 
gun and which set the fuzes. Such a system is almost 
wholly mechanized and automatic, the human ele¬ 
ment normally entering only at the initial moment 
of choosing the target, and at the final moment of 
firing the gun. In the most completely developed 
systems, even the loading of the gun is largely or 
wholly mechanized. 

The testing or evaluation of an antiaircraft fire 
control system involves a large number of consider¬ 
ations, some of which prove to be involved and 
difficult. The principal points appear to be as follows: 

I. The determination of the errors in components 
of the system, or in the total system, when the inputs 
are without error. 

The component in question might be a single am¬ 
plifying circuit in the radar range equipment, or a 
slide multiplier in the predictor, or the fuze timing 
mechanism; or a larger component such as the whole 
predictor, or the gun servo. 

An important part of this error (namely that part 


which would remain if all mechanization were per¬ 
fect), can often be analyzed and computed mathe¬ 
matically, whether or not the physical equipment 
exists. Many studies of this sort were carried out 
by Division 7, and some by AMP. 

But actual test of physical equipment is also es¬ 
sential, for even with perfect inputs, real equipment 
fails to perform exactly in accordance with the cal¬ 
culated theory, there being certain inevitable errors 
due to imperfection of parts, friction, spring of parts, 
wear, uncompensated temperature effects, etc. 

Experiment is essential in the determination of 
these errors for another reason — namely that certain 
phenomena are not amenable (at least at this stage 
of affairs) to reliable predictive analysis. Thus the 
fragmentation characteristics of a certain shell must 
be determined experimentally. (See Chapter 11.) 

II. The determination of input errors , and the 
determination of the effect of these input errors on the 
output performance of components of a system, or of 
the whole system. 

At this point is introduced the vitally important 
matter of the alertness, accuracy, and stamina of the 
human operating personnel, as these factors are in¬ 
fluenced by selection and training, and by operating 
conditions. The design of the system should take 
intimate account of the ability and limitation of the 
personnel. For example, the smoothing of data which 
accompanies the prediction process must be based 
directly on a knowledge of the fluctuations due to 
tracking errors, and an increase in theoretical pre¬ 
cision of a device may be more than cancelled out by 
increased difficulty of operation. This is a field in 
which Division 7 has done much pioneer and funda¬ 
mental work. 

It is clearly important that all such tests be carried 
out under as realistic conditions as possible, that the 
testing be quantitative and objective, and that the 
tests be adequately planned and reliably interpreted 
by use of statistical techniques. Here again Division 
7 has set new standards, notably through the de¬ 
velopment of dynamic testers for prediction, the de¬ 
velopment of great testing engines for aircraft fire 
control equipment, and the development of exten¬ 
sive and specialized computing equipment. The 
AMP’s contributions have been minor and inci¬ 
dental. 

III. The determination of the overall effectiveness 
of the fire control system. 

The ultimate test of an antiaircraft defense is its 
ability to shoot down enemy planes. Thus it is natural 


CONFIDENTIAL 



INTRODUCTORY REMARKS 


147 


to attempt to evaluate, for any given system, some 
overall index such as “probable rounds per bird,” 
assuming a certain type of target and a certain course 
of target relative to gun. 

Such calculations involve, first of all, a knowledge 
of the pattern of error of the actual burst points in 
the sky relative to the desired burst location. This 
knowledge comes primarily from the studies and tests 
outlined above under I and II. It is describable in 
terms of the error in the mean point of impact of 
many bursts, this error in the mean point of impact 
normally changing its value as the target moves along 
its course. In addition to this slowly changing bias 
in the location of burst points, there is a random 
error of the individual bursts with respect to the 
mean point of impact, and under certain circum¬ 
stances a correlation of greater or lesser importance 
between the position of individual succeeding shots. 
In the case of automatic weapons firing in bursts, it 
is sometimes convenient to speak of bias errors which 
are steady during the burst, but whose value would 
fluctuate from burst to burst, and might change 
systematically from point to point on the target 
course. 

These studies of effectiveness must also deal with 
the whole complicated business of terminal ballistics 
— the size and weight distributions of shell frag¬ 
ments, the angular distribution of fragments, the ve¬ 
locity distribution of fragments, and the fall-off law 
for the fragment velocities as this depends on alti¬ 
tude. The vulnerability of the target plane must also 
be carefully considered, studying the probable effect 
of single or multiple hits of fragments of various 
characteristics on all the aircraft components. The 
orientation of the target is involved, as well as the 
relative velocity of target and shell. 

It is also important to question how the effective¬ 
ness of the antiaircraft fire control system is affected 
if the target plane follows a course (curved flight or 
evasive action) which was not contemplated in the 
design of the predicting mechanism. 

And “trial fire” procedures, intended to furnish 
optimum empirical values for certain correction 
parameters (such as ballistic wind, muzzle velocity, 
air density), have an important influence on overall 
effectiveness. 

Relative to the items which come under this head¬ 
ing, AMP carried out a considerable amount of work 
which will be briefly summarized or referred to in 
this chapter and in Chapter 11. In this chapter, Sec¬ 
tion 10.1 is introductory. Sections 10.2 and 10.3 refer 


to dry run errors in tests of fire control systems 
without shooting. Section 10.4 is a theoretical com¬ 
parison of a linear and a quadratic predictor. Sec¬ 
tion 10.5 gives estimates of the contributing errors of 
a particular fire control system. Section 10.6 dis¬ 
cusses two nonlinear variants of a conventional linear 
prediction circuit. Section 10.7 discusses trial fire 
procedures in the case of a projectile with preset fuze. 
Sections 10.8, 10.9, and 10.10 are brief discussions of 
special topics involving antiaircraft equipment with 
which AMP was concerned. In Chapter 11, AMP’s 
analytic studies of fragmentation and damage are 
reported, together with certain pioneering British 
and Section T papers. 

lo.i.i Fire Control Systems 

Consider a target aircraft in motion along its 
course. Suppose that a shot is to be fired at the in¬ 
stant at which the target is at a point T i of its course, 
called the present position. If the motion of the target 
in the instants following the firing instant is con¬ 
sidered as known, there is then determined the future 
position T 2 on the course, namely, that point on the 
course at which the projectile should be aimed in 
order that the shot be most likely to hit the target. 
Thus the pointing of the gun (and the fuze setting, in 
the case of a preset fuze) should be such as would 
produce a hit at the future position T 2 , according to 
the ballistic tables. 



Figure 1. Position of gun and target. 


The predictor is that part of the fire control system 
which determines the gun settings which are appro¬ 
priate for each shot. It is clear that the motion of 
the target in an interval preceding the firing instant 
must influence the solution produced by the pre¬ 
dictor, for such prior motion of the target is the only 
available source of information about the motion of 
the target during the time of flight of the projectile. 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


It is also clear that the solution produced by the pre¬ 
dictor cannot be correct in all cases, since the target 
may move in different ways during the time of flight 
of the projectile, and, in certain cases, will be able to 
avoid damage by the projectile directed at a future 
position T 2 appropriate for a course different from 
the one actually flown after 7\. 

Thus the operation of the predictor includes an 
extrapolation of the past motion of the target into 
the future. 

The way in which the motion of the target prior 
to the instant of prediction is fed into the predictor is 
as follows. Range to the target at each instant is 
measured by radar or optically. The angular direc¬ 
tion to the target at each instant is established by 
moving a line (the axis of a telescope) in the predictor 
so as to point at the target continuously. This process 
is known as tracking. Thus, except for the ranging 
and tracking errors, the present position of the target 
(relative to the predictor platform) is an input to the 
predictor at each instant. 

In addition to taking account of the motion of the 
target, the predictor must make use of ballistic in¬ 
formation of the sort contained in the firing tables for 
the gun and projectile. In particular, the appropriate 
pointing of the gun will not usually be toward the 
future position; it will be such as to achieve a hit on 
the future position. 

In the case of heavy antiaircraft, the input to the 
predictor will include the ballistic wind, the ballistic 
temperature, the ballistic air density and an estimate 
of the muzzle velocity, and perhaps other settings. 

The mechanism which transmits the angular gun 
orders (that is, the desired gun settings which refer 
to the pointing of the gun barrel) from the predictor 
to the gun operates so rapidly that one may consider 
the transmission as effectively instantaneous. The 
calculation of the angular gun orders by the predictor 
is therefore based on the presumption that the gun 
is fired at the very instant of prediction. In the case 
of a preset fuze, however, the fuze must be set before 
the shell is placed in the breech of the gun and fired. 
The interval of time between the instant at which the 
predictor computes the fuze setting and the instant 
at which the projectile is fired is the fuze dead time. It 
must be known in advance and be fed into the pre¬ 
dictor. 

Considering the entire fire control system, one may 
say that its input consists of the motions of the target 
and the gun and predictor platform, the tracking in¬ 
put, the ranging input, a number of settings related 


to wind, air density, temperature, muzzle velocity 
and the like, and, in the case of a preset fuze, the fuze 
dead time. 

The output of the fire control system is the se¬ 
quence of gun settings effected by the gun and the 
resultant trajectories. The gun settings are the 
angular gun settings which determine the direction of 
the rifle of the gun, and in the case of a preset fuze, 
the fuze setting. 

The behavior of a particular projectile is not de¬ 
termined exclusively by the gun settings at its firing; 
there is superposed on these gun settings a dispersion 
or random variation: The bursts or trajectories, ob¬ 
tained from a series of shots at precisely the same gun 
settings, will be distributed about their average posi¬ 
tion (which is given in the ballistic table). In the 
case of proximity and contact fuzes, the detonation 
of the projectile is determined in part by the target 
itself. 

A general (although incomplete) discussion of the 
character of antiaircraft weapons, heavy antiaircraft 
predictors, fire control equipment for light antiair¬ 
craft, and antiaircraft radar equipment as well as 
the reported operating characteristics of a number of 
specific weapons is given in a memorandum. 6 

10.1.2 Time Base of Predictor 

Because the prediction is in part an extrapolation 
of the prior motion of the target as observed, the 
effect of errors of observation such as ranging and 
tracking errors would be serious if the observations 
were not averaged over a sufficiently long time in¬ 
terval. 

To illustrate this point, consider a predictor which 
extrapolates in the following linear fashion: The 
course of the target during the time of flight is pre¬ 
sumed to be along the straight line joining the present 
position Ti of the tdrget with the position T 0 of the 
target C seconds before the present instant, where 
C is some constant known as the time base of the pre¬ 
dictor. Suppose the target is indeed flying at a con¬ 
stant velocity (that is, in a straight line at a constant 
speed), but suppose that the input to the computing 
mechanism of the predictor corresponds to a wavy 
motion which deviates from being straight because 
of input and machine errors. Then the extrapolation 
may be grievously wrong, if the time base is small 
compared to the time of flight, as is indicated in 
Figure 2. This observation is related to the fact that 
although an error may be small its rate of change 


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INTRODUCTORY REMARKS 


149 


may be large; and the rate of change of the observed 
target position affects the extrapolation to the future 
position. 

FUTURE POSITION (LONG TIME BASE) 



Figure 2. Long vs short time base. 

Thus, other things being equal, a long time base 
improves prediction. Antiaircraft engagements are of 
limited duration and it is desirable to achieve effec¬ 
tive shooting as early in the engagement as is pos¬ 
sible. For this reason, a short time base would be 
more advantageous than a long one, if there were 
not compensating disadvantages. 

10.1.3 Assessments of Fire Control 
Systems 

Practically all the work of AMP on antiaircraft 
had to do with the assessment of fire control systems 
in the following sense: The AMP work was an ap¬ 
praisal of the effectiveness of a fire control system 
or of one of its parts, either by observation of per¬ 
formance in the field or in a laboratory, or by the 
study of its design. 

An empirical method of assessing a fire control 
system is to set it up against the enemy and to count 
the number of planes shot down and the number of 
rounds fired or the number of engagements. Assess¬ 
ments of this sort apply specifically to the inputs 
which occurred in the cases observed. Accordingly, a 
comparison of two fire control systems on the basis 
of such observations should be tempered by an 
awareness of the extent, if any, to which the actual 
input tended to favor or disfavor one system against 
the other. For example, one system may be superior 
to another against targets taking evasive action. 
Such a superiority will not be in evidence if the enemy 
flies only straight courses in the observed engage¬ 
ments. 

An assessment of a fire control system, of the sort 
just described, in the field against the enemy should 
be the last of a series of assessments carried out all 
during the interval from the initial design of the 


fire control system to its production and use in 
quantity. For, if these earlier assessments are not 
carried out, adjustments and improvements of the 
fire control system would be either delayed or lost 
altogether. 

One may consider a series of types of assessment, 
of which the one just discussed is at the extreme, 
since it is an observation of the fire control system 
in action in which the inputs are subject to a mini¬ 
mum of control on the part of the observer. At the 
other extreme from the assessment just described is 
one which consists entirely of paper work. The com¬ 
ponent parts of the fire control system are described 
mathematically; the output is then determined by 
calculation, and appraised. Fire control systems are 
so complicated that a certain amount of idealization 
has been necessary in assessments of this type. Ac¬ 
cordingly, the validity of the results depends upon 
the extent to which the idealizations approximate 
the essential elements in the situation. Experimental 
checks of the entire assessment or of parts of the 
assessment can establish the soundness of a theoreti¬ 
cal assessment and permit one to have confidence 
in its conclusions. 

One advantage of an assessment which is entirely 
theoretical is this: Initial conditions can be varied 
readily and the whole range of appropriate inputs can 
be studied. Thus the scope of the analysis is sub¬ 
stantially broader than can be achieved by observing 
the fire control system in action. For example, if one 
wishes to consider the behavior of the fire control 
system under U\ types of tracking, n 2 types of rang¬ 
ing, n 3 types of target motion, n 4 types of predictor, 

• • •, one must consider in all n\ntfi$n± • • • situations. 
The magnitude of the number of cases soon becomes 
prohibitive. Nonetheless, the theoretical analysis can 
be carried out, and a moderate number of experi¬ 
ments can be judiciously chosen and used as checks 
of the theory. Judicious and properly planned ex¬ 
perimental investigations combined with theoretical 
analyses provide a powerful method of research. 

Any experimental assessment should be as realistic 
as possible. This implies that it should involve shoot¬ 
ing, provided the attainable motion of the target and 
other conditions are representative of actual condi¬ 
tions. As the maneuverability of tow targets is 
definitely limited, there will be situations in which a 
camera test against fast moving real airplanes, pos¬ 
sibly flying evasive courses, will be a more realistic 
simulation of combat conditions than is attainable 
with shooting. In a camera test, everything but the 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


shooting may be simulated experimentally. The effect 
of shooting is then superposed on the experimental 
observations. 

For the most part, the theoretical analyses of anti¬ 
aircraft equipment conducted by the AMP applied 
to parts of the fire control system, rather than the 
whole. 

10.1.4 Calculation of Probabilities 

In any assessment, experimental or theoretical, it 
is desirable to calculate the probability of destroying 
the target during the engagement or the probability 
of hitting the target with one shot, or some similar 
probability. (The first probability is an estimate of 
the proportion of engagements in which the target 
would be destroyed. The second is an estimate of the 
proportion of shots which would hit the target.) 
These probabilities are a better measure of the per¬ 
formance of the fire control system than is a descrip¬ 
tion of the amounts by which the hypothetical pro¬ 
jectiles miss the target. For the probabilities are a 
measure of precisely that effectiveness for which the 
fire control system is intended. It would be proper to 
stop with a description of the misses only if such a 
description were, in effect, a description of the above 
probabilities. Now the process of converting the 
misses into probabilities involves the angular target 
size (which varies throughout an engagement) and 
the gun dispersion; the probability of hitting the 
target or of destroying the target is not at all propor¬ 
tional to the amount of the miss. The vulnerability of 
the target is sometimes complicated. 

Thus the step from the amounts by which the 
hypothetical projectiles miss the target to the proba¬ 
bilities is essentially complex and often cannot be 
accomplished by an intuition based on a brief de¬ 
scription of the amounts of the misses alone. It would 
therefore seem worth while to compute the proba¬ 
bilities whenever the gun errors are sufficiently re¬ 
liable. There will be cases in which a rough computa¬ 
tion of the probabilities will be sufficient. This matter 
is discussed further in Section 10.2. 

10.2 ANALYSIS OF DRY RUN ERRORS 

Suppose that a series of trials is carried out (in 
reality or perhaps only conceptually) in which all the 
elements of the fire control system are involved with 
one exception: There is no shooting; instead a record 
of the gun settings is taken, photographically^ or 


otherwise. Each trial is a simulation of an engage¬ 
ment of the fire control system against an enemy 
plane. Each trial will be called a dry run. 

The dry run error at any firing instant may be de¬ 
fined as the displacement from the target to the 
average position of a projectile fired at the firing 
instant with the gun settings that were in effect in 
the dry run, the displacement being measured at the 
instant at which a correctly aimed projectile with 
zero dispersion would hit the target. In the case of a 
proximity or contact fuze the displacement is meas¬ 
ured perpendicular to the average trajectory (rela¬ 
tive to the target); in the case of a preset fuze it is 
measured as a vector in 3-dimensional space. An indi¬ 
cation of a method of calculating dry run errors, to¬ 
gether with explicit definitions, is given in Section 
10.3. 

The present section will consider the analysis of 
records of dry run errors. The concepts and methods 
discussed are pertinent to antiaircraft problems (par¬ 
ticularly those which are concerned with automatic 
weapons) as well as to aerial gunnery and rocketry 
problems. The work of AMP along the line indicated 
was principally (but not entirely) in connection with 
air-to-air gunnery. 

We conclude this introduction with a remark about 
the planning of the dry runs. The performance of a 
fire control system in an actual situation is the result 
of a large number of component causes, some of 
which are: 

1. The type of engagement (that is, the particular 
motions of the centers of mass of the target and the 
gun platform). 

2. The yaw, pitch, and roll of the gun platform, 
especially important in naval engagements and in 
air-to-air gunnery. 

3. The tracking errors. 

4. The ranging errors. 

5. Atmospheric conditions, accuracy of adjust¬ 
ment, degree of wear, the particular harmonization, 
and the like, of the system. 

Consider an investigation whose object is to de¬ 
termine the average or expected behavior of the 
fire control system when some or all of the conditions 
mentioned above (1 to 5) vary. In certain investiga¬ 
tions one may wish to restrict as many of the con¬ 
ditions as possible, and study the effect of the vari¬ 
ations of the others. For example, in an experimental 
investigation, one could specify that the target fly a 
particular course relative to the gun, and that there 
be no yaw, pitch, and roll of the gun. In all investi- 


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ANALYSIS OF DRY RUN ERRORS 


151 


gations, there should be an adequate representation 
of the different combinations of the conditions (1 to 
5) which are to be varied, and, in so far as it is 
possible, the weights given to the different conditions 
that are considered should correspond to the fre¬ 
quency with which the conditions are expected to 
occur in reality. 

10.2.1 Superposition of Firing 

Effects 

The effects of firing are adjoined to the dry run 
errors by means of the theory of probabilities. The 
dry run error refers to the behavior of an average pro¬ 
jectile; any particular projectile, if it is fired during 
the dry run, would deviate from the average of all 
projectiles that might be fired at the same instant, 
in accordance with the probability distribution func¬ 
tion which describes the behavior of the set of all 
projectiles. The variation from the average to the 
individual will be referred to as the dispersion ; this 
dispersion is to comprehend those effects (and only 
those effects) which would be present with actual 
shooting but which are absent in the dry runs. The 
dispersion is an elusive quantity to determine, but 
there is no reason to suppose that the approxima¬ 
tions that are made are unsatisfactory. 

Formulas will be given in Section 10.2.6 which will 
permit the calculation of the probability p that a 
particular shot at a particular instant of a dry run 
would destroy the hypothetical target. The formulas 
themselves are less important than the way in which 
the quantities p are used once they have been de¬ 
termined. 

The meaning of the probability p is as follows: It is 
the proportion of shots which, fired under the condi¬ 
tions which prevailed in the dry run at the firing 
instant against a target which moved before and 
after the firing instant in precisely the way that the 
target did move, would have destroyed the target 
supposing that many such shots were fired. 

The probability p will depend on the dry run 
error at the instant in question, the vulnerability of 
the particular target to the ammunition, and the 
dispersion. 

10.2.2 Unconditional Vulnerability 

A target is said to be unconditionally vulnerable if 
the probability that any shot will destroy it is inde¬ 
pendent of any damage (other than complete de¬ 
struction) that the target may previously have sus¬ 


tained. Thus, an unconditionally vulnerable target is 
destroyed, if it is destroyed at all, by the effects of 
one shot. Although actual targets are probably not 
unconditionally vulnerable in the strict sense with 
respect to any particular kind of ammunition, never¬ 
theless, the idealization that they are unconditionally 
vulnerable is useful, and at least in many cases, gives 
results which are essentially correct. 

One reason for this is as follows: The probability 
that two or more projectiles will produce hits on 
mutually supplementary parts and thereby destroy 
the target, when each projectile by itself would fail to 
destroy the target, is small compared to the proba¬ 
bility that one projectile will destroy. The presented 
area of the component parts which are multiply but 
not singly vulnerable is small, and the probability 
per unit area of obtaining precisely the required set of 
hits is small. 

Moving pictures of a target being downed by a 
stream of caliber 0.50 bullets or a mass of fragments 
of a shell may give the impression that the target is 
downed by the combined effect of many hits. But 
this is not an inescapable conclusion. There is usually 
a multiplicity of hits with each kill of an uncon¬ 
ditionally vulnerable target for several reasons: 
(1) The effect of the lethal hit may be delayed. Con¬ 
sequently all hits occurring after the lethal hit but 
before the destruction appear to contribute to the 
destruction. (2) The probability that a fragment is 
lethal is less than the probability that a fragment 
will hit the target. Consquently, many non-lethal 
hits occur for each lethal one, on the average. 

The reader is referred to Appendix 1 of a report 21 
for a discussion of one aspect of this subject, that is, 
the vulnerability of a fighter aircraft to machine gun 
fire. 

A conditionally vulnerable target is one for which 
there is an appreciable probability that two or more 
projectiles will together destroy the target, each one 
of itself failing to destroy. A target which cannot be 
destroyed but which can be damaged is the opposite 
of an unconditionally vulnerable target. An extreme 
example of the former is the City of London and of 
the latter a compact ammunition dump. 

10.2.3 Survival Probability 

Henceforth it is assumed that the analysis is di¬ 
rected toward a determination of the effectiveness 
of the fire control system against an unconditionally 
vulnerable target. 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


Suppose that shots are imagined during a dry run 
and spaced so as to conform to the rate of fire of the 
fire control system, and that p h p 2 , • • •, p n are the prob¬ 
abilities defined in Section 10.2.1, that the respective 
shots would, of themselves, destroy the target. Then 
1 — pi, 1 — p 2 , • * •, 1 — p n are the respective proba¬ 
bilities that the shots will fail to destroy the target. 
Accordingly, the survival probability for the n shots is 

S= (1 Pi) (1 -P*) (1 ~Pn), (1) 

since the target is unconditionally vulnerable. The 
destruction probability is 

d = 1 — s. 

The probabilities s and d are the proportions of en¬ 
gagements in which the target would survive or be 
destroyed, respectively, it being supposed that the 
entire dry run is repeated exactly many times with 
shooting each time. Thus it is presumed that the pilot 
of the target in combat would not take any special 
evasive action (other than that present during the 
dry runs) as the result of the effects of the shots early 
in the engagement. 

The probability s as given by equation (1) is too 
large, if the target is conditionally vulnerable. 

10.2.4 Expected Values — Measures 
of Performance of Fire 
Control System 

The probabilities d, s, and p apply to a particular 
run or a particular instant of a particular run. For 
this reason they are not measures of the entire per¬ 
formance of the fire control system. Such measures 
are rather the average or expected values of d, s, and 
p over the set of all real runs. We shall use capital 
letters to denote estimates of the expected values of the 
corresponding small letters. Thus, D, S, and P are 
estimates of the expected values of d, s, and p, re¬ 
spectively. 

The quantity D is an estimate of the proportion of 
engagements in which the target would be destroyed. 
It is, therefore, a sound measure of the performance 
of the fire control system. It takes account of the dis¬ 
persion of the projectile, the vulnerability and size of 
the target, and the serial correlation of the dry run 
errors along each course. (The serial correlation is 
discussed in the next section.) 

The quantity P is an estimate of the proportion of 
shots which would destroy the target. It may be a 
sound measure of the performance of the fire control 


system, even though it does not take account of the 
serial correlation. 

Let us denote by E an estimate of the expected 
value of the radial dry run errors. The quantity E is 
a measure of the performance of the fire control sys¬ 
tem which does not take account of projectile dis¬ 
persion, target vulnerability, target size, or serial 
correlation. It may, nonetheless, be adequate in 
certain cases. 

Suppose that each gun of a fire control system is 
replaced by two guns and that this replacement does 
not change the dry run errors. Then the survival 
probability s will be replaced by its square s 2 . This 
follows directly from formula (1). It has been sug¬ 
gested 30 that 

c = —log s 

be a measure, the comparison index , of the perform¬ 
ance of a fire control system. The comparison index 
c has the following properties. (1) The greater c is, 
the greater the probability of destroying the target. 
(2) Replacing each gun of the system by two guns 
(subject to the same dry run error) has the effect of 
doubling c. Properties (1) and (2) hold also for C, the 
expected value of c over all runs. 

To compare the meaning of C with that of D, one 
might note that knocking down twice as many enemy 
planes in the same number of engagements has the 
effect of doubling D. 

10 . 2.5 Serial Correlation of Dry 
Run Errors 

The dry run errors at the firing instants of a run 
are uncorrelated (statistically independent) if they have 
the following property: The error at an instant is 
statistically independent of the errors at earlier in¬ 
stants so that a knowledge of the errors at the earlier 
instants would not help a person to predict the error 
at the later instant. 

It is clear that the dry run errors are correlated 
since the causes of the errors at different instants are, 
in part, the same or related. However, if it were true 
that the errors were uncorrelated, the calculation of 
S, an estimate of the expected survival probability, 
would be considerably simplified. This is because the 
expected value of a product of statistically inde¬ 
pendent factors is the product of the expected values. 
Accordingly, equation (1) would imply that the ex¬ 
pected value of s is 

(1 - Pl) (1 "Pn), 


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ANALYSIS OF DRY RUN ERRORS 


153 


when P i, • • •, Pn are respectively the expected 
values of pi, • • •, p n . The effect of serial correlation 
will be important to the extent to which the above 
product fails to approximate the expected value of s. 
Among other things, rate of fire will certainly affect 
the importance of the serial correlation. 

One may illustrate the effect of the correlation of 
dry run errors by the following examples: 

Illustration 1. Consider two fire control systems A and B. 
Suppose that system A is half on and half off the target during 
each run, and that system B is entirely on the target during 
half the runs and entirely off during the other half. Assume 
that bullet dispersion is zero and that one hit on the target 
would destroy it. Assume (as we do throughout) that the 
fire control system itself is invulnerable. Gun system A will 
then destroy the target in all engagements, and system B 
will destroy the target in only half of the engagements. 
Nevertheless, the probability that one shot will destroy the 
target is the same for both systems. For these two systems, 

Pa = Pb = h; D a = 1 ; D b = l 


Thus, if the quantity (n — 1/2)2(1 — 0) is sufficiently 
small, the D criterion will bp equivalent to the P cri¬ 
terion. If the quantity is large, the criteria will be 
essentially different. 

Illustration 2 ig an extreme case in the sense that 
system A contains no serial correlation at all, and 
system B contains the ultimate of positive serial cor¬ 
relation: If one error is zero, all others are zero ; and 
if one error is different from zero, all are very large. 
It is possible that real fire control systems exist 
which contain negative correlation, that is, in which 
there will be a tendency for the system to get on the 
target when it is off, or vice versa. It might be con¬ 
jectured, however, that the P criterion will be ade¬ 
quate whenever it is adequate in the Illustration 2, 
for n, 0, and z that are appropriate. 

It has been pointed out 26 that there are cases in 
which serial correlation will increase the probability, 
contrary to its effect in Illustrations 1 and 2. 


It has been conjectured that the contrast in Illus¬ 
tration 1 is partly due to the absence of dispersion, 
and that a comparison of systems on the basis of the 
quantities D and P would give essentially the same 
results when the dispersion is large compared to the 
target. 

The reader may get some insight into this question 
by considering the following modification of Illus¬ 
tration 1. 

Illustration 2. Consider two fire control systems A and B. 
Let 0 be any positive number less than unity. 

Suppose that system A is, on the average, squarely on the 
target during a 0th part and badly off the target during a 
( 1 — 0)th part of each run, all dry run errors being statistically 
independent. 

Suppose that system B is either squarely on or badly off the 
target during the entire run, and suppose that the first possi¬ 
bility occurs, on the average, for a 0th part of all the runs, and 
the second, for a (1 — 0)th part. 

Taking into account target size, vulnerability, and disper¬ 
sion, let z be the probability that a single shot will destroy 
the target when the error is zero. Suppose that n shots are 
fired, and that the angular target size is taken as constant 
for all n shots. 

Then the two systems have equal expected single shot 
probabilities, 

P = 02. 

However, the expected destruction probabilities for the two 
systems are (omitting details), 

D a = 1 - (1 - 02 ) w , 

D b = 0 [1 - (1 -*)"} 

Accordingly, 

^ 2(1 — 0) + higher powers of 2 . 


Da 

D b 


1 + 


10.2.6 Calculation of p, Probability 
That a Shot with Given Dry Run 
Error Will Destroy Target 


Let us consider the case of a contact or proximity 
fuze. Let ( x,y) be coordinates with origin at the 
center of vulnerability of the target in a plane per¬ 
pendicular to the trajectory of the average perfectly 
aimed projectile (see Figure 3). Strictly speaking, the 
trajectory is to be taken relative to the target, that 
is, the trajectory is to be the motion of the projectile 
as viewed from the target. The reason for this is that 
the effect of the hits on the target is due in part to 
the velocity of the hitting particles, relative to the 
target. 

Consider an instant at which the dry run error is 
(a, b). It is generally assumed that the actual errors 
(x,y) of trajectories of real projectiles are distributed 
normally about (a, b), that is, that there are quanti¬ 
ties M11, M22, M12 such that g(x,y)dxdy is the probability 
that the shooting error will lie in a rectangle of sides 
dx,dy parallel to the axes about ( x,y ), where 


g(x,y) = 


2tt \/ A 


■Q /2 


Q = -O22O - a) 2 - 2 /Z 12 O — a) (y — b) 
A 


+ mu (y - &) 2 ]> 


M11M22 — Ml2- 


Here mu and M 22 are called the variances of x and y , 
respectively, and M12, the covariance of x and y. These 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


quantities are presumed known. In many cases, the 
pattern is taken as circularly symmetric, that is, 
Mu = M22 = <r 2 and M12 = 0, where a, the standard 
deviation , is a known quantity. Ballistic tables should 
give values of the ps, at least in the case of a fixed 
target and fixed gun. When the target motion is im¬ 
portant, tables describing the behavior of a proximity 
fuze should give appropriate values of the m’s. In the 
case of a contact fuze, the target motion is probably 
unimportant. The variances and covariances are dis¬ 
cussed further in Section 10.3.2. 

At a particular instant of a run, the dry run error 
will not be known exactly. Rather {a,b) will be the 
dry run error contaminated by a measurement error, 
if actual observations are involved in the determina¬ 
tion of (a, b). The measurement errors can be studied. 
It will be assumed that the measurement errors are un¬ 
biased (have mean (0,0)) and have known variances 
and covariance Mn.m, P22,m, and Mi2,m, respectively, 
then it can be shown that instead of the normal 
function g{x,y) given above, one should use the 
normal function g*(x,y), obtained by replacing the 
variances and covariance of g(x,y) by 

Mil — Mil,mi M 22 — M 22 ,m) Ml 2 — Ml 2 ,wi) 

respectively, provided the latter differences are non¬ 
negative. 

The above subtraction of the measurement vari¬ 
ances and covariance is discussed in two reports. 21 * 1,25 
That the subtraction is reasonable can be seen as 
follows. Consider a hypothetical case in which the 
measurement variances and covariance precisely 
equal the bullet variances and covariance, re¬ 
spectively. In such a case, the measurement effects 
which are present in the recorded dry run errors are 
statistically precisely those which would be present 
in a shooting test. In the latter test, the bullet errors 
would replace the measurement errors. Under our 
hypothesis, the two are equivalent. Accordingly, no 
further dispersion should be added to that already 
present due to measurement errors; the variances and 
covariance should be zero. 

Let 8(x,y) be the probability that a projectile with 
error (x t y) will destroy the target. Since the origin is 
the center of vulnerability of the target, 8{x,y) will 
vanish when ( x,y ) is sufficiently far from the origin. 

The probability that a shot at an instant at which 
the recorded dry run error is (a,6) will destroy the 
target should be taken as 

V = f f g*(x,y)S(x,y)dxdy. (2) 

J — 00 J — 00 


If the function 8 is, except for a constant factor, 
itself normal, that is, of the form of the function g, 
the above integration can be carried out explicitly. 
In any case, the function 8 can be approximated by a 
constant times a normal function. In this way it can 
be shown that the probability of destroying the target 
with a single shot is approximately equal to a constant 
multiple of the value, at the center of vulnerability of the 
target, of the normal probability density with mean at 
the center of the bullet pattern and with variances and 
covariance equal respectively to those of the vulnerability 
distribution plus those of the bullet pattern less those of 
the measurement errors. The variances and covariance 
of the vulnerability distribution referred to are de¬ 
fined as follows. 

Suppose that 



8(x,y)dxdy 


c. 


Since the origin of coordinates is at the center of vul¬ 
nerability of the target, 



x8(x,y)dxdy = 



y8{x,y)dxdy = 0. 


The variances and covariance of the vulnerability 
distribution are 


1 f* 00 f * 00 

Mn,® = - I I x 2 8{x,y)dxdy, 

C J — oo J — oo 
^ r* oo r oo 

Mi 2 ,» = - I I xy8{x,y)dxdy, 

c J — 00 — CO 

\ f* CO /» oo 

P 22 ,v = ~ | I y 2 8(x,y)dxdy. 

C J — oo «/ — oo 


The italicized assertion above is equivalent to the 
following: The probability of destroying the target is 
approximately c times the value of g at x = 0, y = 0, 
with Mu? M22) M12 replaced by mu ~b Mn,® — Pn,m', 
M22 + M22 ,v — P22,m) M12 + Pi2,v — P\2,m) respectively. 
For a proof of the assertion, the reader is referred to 
an AMP report. 153 

The Applied Physics Laboratory of Johns Hopkins 
University has found that the probability that a shot 
will destroy the target is approximately 


ka 2 

a 2 + 2s 2 


e _AVa 2 + 2i2 


= (kird 2 ) • 


1 



e -hyi2(s+a*;2)l 


where k and a are constants (operability and radius 
of lethal area) appropriate to the particular fuze and 
target, s 2 is a variance, and h is the radial dry run 
error. 303 This formula is of the type that we have just 


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ANALYSIS OF DRY RUN ERRORS 


155 


discussed, except that the measurement errors are 
excluded, probably as being of small influence. 

10.2.7 Procedures for Estimating 
Expected Survival Probability 

What may be termed the direct procedure for esti¬ 
mating the expected survival probability has been 
studied. 2130 The procedure is to calculate the survival 
probability s for each dry run according to the 
formula (1) and to take as S, the estimate of the ex¬ 
pected survival probability, the average of the s’s for 
the individual runs. If desired, the runs can be 
given different weights in the averaging. The estimate 
D of the expected destruction probability is then 
1 - S. 

The direct procedure has the advantage that no 
special hypothesis is made as to the nature of the dry 
run errors ; they are taken as observed. It has the ad¬ 
vantage, also, that the calculation is fairly simple. 
It has the disadvantage that the calculation appears 
to possess great intrinsic variability. If the dry run 
errors constitute a normal universe (of 2n dimen¬ 
sions), the procedure is statistically inefficient, that 
is, extracts from the data a less reliable conclusion 
than one to be described below. 

It should perhaps be noted that statistical effi¬ 
ciency at the expense of experimental or mathe¬ 
matical simplicity is not necessarily a good thing. 
Overall efficiency and soundness are desired for the 
investigation. It may be easier to fire four times as 
many projectiles or photograph four times as many 
dry runs and then execute relatively simple calcula¬ 
tions than it would be to carry out a complex investi¬ 
gation of the results of fewer runs. It has been 
pointed out 29 that a procedure which is statistically 
efficient for a normal universe may be definitely in¬ 
efficient for a universe which, in some sense, is close 
to being normal. 

The direct procedure has been carried out in a 
number of investigations. Numerical results are given 
in two reports. 16 - 17 

Another report 30 also presents two alternative 
methods for estimating the expected survival proba¬ 
bility. These probably possess less intrinsic vari¬ 
ability than is present in the direct procedure. The 
methods are based on what may be called a hy¬ 
pothesis of quasi-steady error: that the dry run error 
is the sum of two components, the first of which is 
the same at the same instant of all runs considered 
and is called the error in mean point of impact or the 


quasi-steady error; and the second of which is a 
fluctuating error which possesses no serial correla¬ 
tion from instant to instant along a run. The quasi¬ 
steady error might be assumed constant throughout 
straight runs, or it might be obtained by a smoothing 
of recorded dry run errors. 

The hypothesis of quasi-steady error imposes on 
the dry run errors a particular type of correlation. 
Shots at different instants are related in that each 
has a quasi-steady contribution. These contributions 
are functionally related; in other respects the shots 
are independent. It is not known whether the hy¬ 
pothesis of quasi-steady error is a useful idealization 
of the actual behavior of dry run errors or not. 

The calculation of the expected survival proba¬ 
bility is described 31 in the case in which the func¬ 
tional and statistical characteristics of the dry run 
errors are known and in which the statistical char¬ 
acteristics are those of a normal universe of 2 n di¬ 
mensional vectors, where n is the number of shots. 
When the statistical characteristics are indeed of this 
sort and are known, the calculation described is the 
most desirable from the point of view of statistical 
efficiency. A procedure which analyzes the dry run 
errors, estimates the variance-covariance matrix of 
their presumed normal universe of 2 n dimensions, 
and then estimates the survival probability by the 
analysis given in the report 31 would presumably be 
the most efficient procedure statistically, provided 
the universe were indeed of the type assumed. Such 
a calculation would be quite involved, however; 
among other things it would require the evaluation 
of determinants of order 2n or 2n + 1. 

Certain assumptions about the behavior of suc¬ 
cessive dry run errors which would make the compu¬ 
tation of the determinants referred to somewhat 
simpler are suggested. 31 The complexity of the 
method described 31 depends very much on the num¬ 
ber of shots. 

10.2.8 Report of Analysis of Dry 
Run Errors Obtained on 
Dynamic Tester b 

Several variants of the Army M-9 and T-15 di¬ 
rectors used with preset fuzes are considered in a 
report. 13 These were studied in the dynamic tester , a 
device which makes it possible to feed the same air- 

b In this and succeeding sections we shall use the term 
director rather than predictor since all bibliographical refer¬ 
ences adopt this usage. 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


craft course into several directors and to record the 
dry run errors for each director. Only the radial dry 
run errors are used; the directions of the point of aim 
from the true point of aim are not used. The radial 
errors are termed “vector errors” in the report. 13 

The radial errors were recorded at 5 or 1 sec inter¬ 
vals. Several of the courses fed into the tester were 
based on flights of enemy aircraft over England. 

The method of comparing the directors was to 
compare the averages of the single shot probabilities 
p of destroying the target at all the firing instants of 
each run. This method of scoring is sound, if the 
effect of serial correlation is unimportant. 

The observed differences in directors found 13 were 
for the most part small. Among the findings were the 
following: The T-15 with curved flight prediction had 
a significantly higher mean score than the same di¬ 
rector without curvature, although the number of 
courses on which each was better was the same. The 
M9 director, with curvature calculated from the 
second derivative of rectangular coordinates and 
with 10 sec smoothing time for the first derivative 
and 20 sec smoothing time for the second derivative, 
had a significantly higher mean score than the T-15 
director with curved flight prediction. 

10.3 CALCULATION OF DRY RUN ERRORS 

It may be of interest to discuss aspects of the cal¬ 
culation of the dry run errors, since the entire dis¬ 
cussion of Section 10.2 supposed that the dry run 
errors were available to start with. 

10.3.1 Backing-up Process 

Suppose that for each dry run there is available a 
record of the target and gun platform motions and of 
the gun settings throughout the entire engagement. 
One can calculate the dry run errors by backing up 
from impact instants to firing instants in the follow¬ 
ing way. On the basis of the recorded motions, one 
can calculate coordinates (q^q^qi) of the target in a 
coordinate system which is stable during the time of 
flight, that is, which does not roll, pitch, or yaw, and 
which moves, if at all, with a constant known trans¬ 
lational velocity. It is presumed that the center of 
mass of the gun platform moves with a constant 
known velocity. If the gun platform moves with a 
known but varying velocity, the calculation of dry 
run errors is more complicated. Stabilization is dis¬ 


cussed in Section 10.3.3. The coordinates (q\,qz,qz) are 
functions of the clock time t; accordingly, 

qi = qitt), q% = q*(t), q* = qs(t)- 
To compute the dry run error, start with an arbi¬ 
trary instant t = b which is taken to be an instant 
at which a perfectly aimed average shell would hit 
the target. The target would then have coordinates 
qi(b), q2(b) f qs(b). In terms of these coordinates and 
the motion of a stable coordinate system, the ballistic 
tables give the time of flight t f of the perfectly aimed 
projectile to the impact point. Then the firing instant 
for the perfectly aimed projectile must have been 
t = b — t f = a. Next look at the record of gun set¬ 
tings and see what the actual gun settings were at 
t = a. The ballistic tables give the position after tf 
seconds of an average shell fired at t = a with the 
observed gun settings. The displacement to this posi¬ 
tion from the point [qi( 6), <? 2 (6), g 3 (&)] is the dry run 
error for the projectile fired at t = a. This displace¬ 
ment is a vector which shall be denoted by e. (See 
Figure 3.) 



Figure 3. Trajectories as varied from the air mass 
and from the target: Y P , remaining velocity of pro¬ 
jectile; Vt, target velocity; e, dry run error; (these 
vectors are not necessarily coplanar). V = V P — Vt, 
velocity of projectile relative to target; e*, the pro¬ 
jection of e on the plane perpendicular to V. 

10.3.2 Dry Run Error Perpendicular 
to Relative Trajectory 

In the case of a proximity or contact fuze, the 
pertinent dry run error is the projection of the full 
dry run error e (discussed in Section 10.3.1) on the 
plane perpendicular to the relative trajectory. Let 
the projection of e on the plane perpendicular to the 
relative trajectory be the vector e*. Let V be the 


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CALCULATION OF DRY RUN ERRORS 


157 


velocity relative to the target of a perfectly aimed 
average projectile at the impact instant t = b; V is 
the remaining vector velocity of the projectile less 
the vector target velocity. Then, since e • V is the 
projection of e on V multiplied by the length of V, 


as can be seen from Figure 3. This relation permits 
one to obtain the error e* pertinent to a proximity 
or contact fuze from the full error e and the relative 
remaining velocity of the projectile. In many calcu¬ 
lations only the length of the vector e* will be 
needed. This length is 

(e • V ) 2 i 

ee — - 

V ■ V 

Since the time of flight of the properly aimed and 
actually aimed average projectiles are the same, the 
vector e will be approximately perpendicular to the 
line from the gun to the target. 

At the impact instant t = b, the set of all projec¬ 
tiles that might have been fired at t = a precisely as 
aimed in the dry run constitutes a three-dimensional 
pattern about their average position. Their velocities 
relative to the target will differ from V by negligible 
amounts as far as estimating the error is concerned. 
Accordingly, the three-dimensional pattern may be 
projected onto the plane through the target per¬ 
pendicular to V; this gives the two-dimensional pat¬ 
tern that was considered in Section 10.2.7. (This 
question in the case of caliber 0.50 bullets is dis¬ 
cussed in a report. 27 ) 

The variances and covariance u n , ju 22 , and /u 2 of 
the two-dimensional pattern will, strictly speaking, 
depend on e and V. Future studies of vulnerability 
and kinematics would seem appropriate to determine 
average or suitable values of these variances and 
covariance. 

10 . 3.3 Stabilization of Observations 

Made from Rotating Gun Platform 

The calculation of the stabilized coordinates from 
coordinates relative to a rotating gun platform and 
measurements of the rotation of the gun platform is 
a problem which was considered by AMP in the case 
of airborne fire control systems. The methods apply 
to the antiaircraft case and are as follows. 

An absolute stabilization is a determination of target 
coordinates in a system which is unrotated through¬ 


out the entire experiment. A local stabilization , on the 
other hand, is a determination of target coordinates 
in a system which is unrotated throughout the time 
of flight of the projectile, but which is not necessarily 
the same for all projectiles. If the object of the calcu¬ 
lation is to determine the radial dry run error, a local 
stabilization is often sufficient. The reason for this is 
that the radial error is a quantity which, itself, is in¬ 
dependent of coordinate systems. Whether a local 
stabilization is sufficient in a particular case will 
depend, in part, on the deviation of the “up” direc¬ 
tion of the particular stabilized coordinate system 
from the true direction of gravity. 

When a local stabilization is permissible, the ad¬ 
vantages gained from its use are twofold: (1) For the 
two instants, impact time and firing time, only one 
stabilization is necessary. (2) The amount of roll, 
pitch, and yaw during a time of flight is small; hence, 
one may use certain correction formulas which are 
valid for small rotations, but not otherwise. These 
formulas are first order corrections. The process is 
analogous to estimating the increment of a function 
by the increment along its tangent line. Local stabili¬ 
zation is discussed in three reports. 15 ’ 22> 23 

When an absolute stabilization is required, there 
are different procedures of calculation. One may use 
the exact but cumbersome equations of rotation. A 
second method is the use of the gnomonic charts. 15 *’ 19 
A third method involves the following idea: 24 For 
each direction in the celestial sphere, the changes in 
zenith distance and bearing for one degree of roll are 
calculated. The changes for R degrees of roll are then 
approximately R times the changes for one degree. 
This approximation is very satisfactory for rolls up 
to 7°, provided the target elevation is under about 
30°. Actually, the tabular correction for one degree 
of roll is not that correction which would be exactly 
accurate for such a roll, but rather that correction 
which, used in the way described, gives the best re¬ 
sults for rolls between 0° and 7°. 

Essentially the same table is used to correct for 
pitch. After coordinates have been corrected for roll 
and pitch, the correction for yaw is trivial. This pre¬ 
sumes the convention that the change from stabilized 
to unstabilized coordinates is accomplished in the 
order: yaw, pitch, roll. If other conventions are fol¬ 
lowed, the procedure of correction is changed ac¬ 
cordingly. 

The above method is analogous to approximating 
the increment of a function by measuring the incre¬ 
ment along a judiciously chosen chord of the graph. 


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STUDIES OF ANTIAIRCRAFT EQUIPMENT 


This method may be extended to permit one to cor¬ 
rect for rotations up to 15°. The extension involves 
the use of other tables, which would apply to the 
range 7° to 15°. For each zenith and bearing, one such 
table would give a constant value of roll and the cor¬ 
rection to zenith per degree of roll in excess of the 
constant amount given. A similar table would refer 
to bearing. The approximation would again be a 
linear one (a chord of the graph), but the chord 
would not pass through the origin. 

10.4 CURVED FLIGHT DIRECTOR 

This section gives the results of a theoretical com¬ 
parison of a particular linear and a particular quad¬ 
ratic director. It is pertinent to the problem of the 
design of new directors for several reasons: (1) The 
methods used may be suggestive to other investi¬ 
gators. (2) The conclusion that the quadratic director 
would be of marginal value compared to the linear 
director may be of interest. (3) Though particular 
directors are considered, there is no reason to doubt 
that their performance is indicative of the perform¬ 
ance of a wide class of directors. 

The work of AMP in this problem ll - 12 was aided 
by the cordial and effective collaboration of the Bell 
Telephone Laboratories. 

10.4.1 Types of Directors 

Consider a director which operates on the basis of 
rectangular coordinates ( x,y,z ,) of the target: The 
director contains a unit which converts the range and 
the angular coordinates of the target (obtained from 
the ranging and tracking) into rectangular coordi¬ 
nates referred to axes fixed in the director platform. 
(The advantage of such conversion stems from the 
fact that the components x, y, z of target velocity are 
constant, whereas the rates of change of range or 
angles are not normally constant when the target 
moves at a constant velocity. 

The coordinates x,y,z are functions of clock time, t : 

x = x(t), y = y(t), z = z(t). 

At a particular instant t = 1 1, these coordinates have 
been fed into the computing unit for all instants up 
to t\. In order to produce a solution at the instant ti 
the director must have at that instant explicit func¬ 
tions X(t ), Y(t), Z(t) in which values of t later than 
h can be substituted. Thus the extrapolation that 
has been referred to in Section 10.1.2 can be per¬ 
formed by means of the functions X{t) f Y(t), Z(t). 


A large class of directors can be defined as follows. 
(Only the x coordinate will be considered since the y 
and z coordinates are treated similarly.) Let a time 
base c be chosen and a weight function w = w(u) 
defined for u between 0 and c. Let X(t) be that linear 
function of t which minimizes the integral 

C [X(£) — x{tyfw{ti — t)dt (3) 

Jtl-C 

of the square of the deviation from the observed x to 
the explicit X over the interval c seconds before the 
present instant fi, the squared deviation being 
weighted according to the function w as indicated in 
equation (3). It can be shown that there is precisely 
one linear function X which minimizes the integral 
(equation 3). 11 A director which uses this function X 
will be called the linear least square director with time 
base c and weight function w. The quadratic least 
square director with time base c and weight function 
w is defined in exactly the same way except that the 
function X(t) is that quadratic function in t which 
minimizes the integral (equation 3). 

Suppose there were neither errors in the input to 
these directors nor machine errors in the directors. 
Then the prediction of the future position would be 
correct, provided the target motion is such that the 
coordinates x,y,z of the target can be expressed as 
quadratic or linear functions of clock time t, re¬ 
spectively, in the case of the quadratic or linear 
directors. 

As a matter of fact, by varying the time base c and 
the weight function w one may obtain a great variety 
of types of extrapolation: The class of directors that 
has been defined is very wide indeed, and it is the 
conjecture of the present author that the class com¬ 
prehends approximations of all types of extrapolation 
that have been used in directors. 

In the following subsections of Section 10.4, the 
special case of constant weights 

w = 1 

is considered. Any other particular case can be in¬ 
vestigated by the same methods. 

10.4.2 Prediction Errors, Supposing 
Perfect Input, Perfect Operation, 
and Perfect Determination of 
Time of Flight 

The errors in prediction by the linear and quad¬ 
ratic directors for various types of target motion are 


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CURVED FLIGHT DIRECTOR 


159 


The Function L 
Time of flight T = 10 sec 
For a helical course, the error in future position is RL. 

For a sinusoidal course, the error in future position is R(L/\/2). 
L is given in the body of the table. 
c is the time base of the director. 

2irq is the period of the helical or sinusoidal motion of the target. 
R is the radius of the helix or amplitude of the sinusoid. 


Linear Director 


Period 

2irq 

seconds 

\ c 

\ seconds 
Q \ 

seconds \ 

0 

5 

10 

15 

20 

12.6 

2 

6.0 

5.35 

2.8 

0.79 

0.67 

18.8 

3 

4.05 

4.95 

4.7 

3.4 

1.6 

25.1 

4 

2.6 

3.6 

4.1 

4.1 

3.5 

37.7 

6 

1.3 

1.9 

2.4 

2.9 

3.15 

56.5 

9 

0.59 

0.90 

1.2 

1.5 

1.8 

75.4 

12 

0.34 

0.52 

0.71 

0.92 

1.1 

94.2 

15 

0.22 

0.335 

0.47 

0.605 

0.75 


Quadratic Director 


Period 

2irq 

seconds 

\ c 

seconds 

9 \ 
seconds \ 

0 

5 

10 

15 

20 

12.6 

2 

13.2 

18.0 

17.0 

11.0 

2.9 

18.8 

3 

5.0 

8.2 

11.0 

11.0 

9.6 

25.1 

4 

2.3 

4.1 

5.8 

7.25 

8.0 

37.7 

6 

0.73 

1.3 

2.1 

2.9 

3.7 

56.5 

9 

0.22 

0.42 

0.675 

0.98 

1.3 

75.4 

12 

0.095 

0.18 

0.29 

0.43 

0.6 

94.2 

15 

0.049 

0.093 

0.15 

0.23 

0.32 


considered. 12 An analysis is accomplished in the fol¬ 
lowing way: It is first supposed that the time of 
flight T is accurately known. The average (root mean 
square) error in future position is then determined for 
certain specified courses. The average is taken over 
the set of all possible positions of the gun for which 
the time of flight against the target will be precisely 
T seconds. The average error for more general courses 
is then compounded from the errors for the particular 
courses. 

Thus suppose that the target is moving in a helix 
of radius R yards, that it makes a complete revolu¬ 
tion in 2tt q seconds, and that the time base of the 
director is c seconds. The component of target ve¬ 
locity parallel to the axis of the helix and the angular 
velocity about the axis are supposed constant. Then 
if the inputs are entirely correct and if the computing 
mechanism is mechanically perfect, the error in 
future position will be RL, where L depends on q, c, 
T, and n: 

L = L n (q,c,T), 


the index n being equal to 1 for the linear director and 
2 for the quadratic director. 

The formulas for the function L and the results of 
certain computations based on these formulas are 
given. 12 Certain of the computations are cited below. 

The error function L applies also to sinusoidal 
target motions. Suppose that the target moves along 
a sine curve of amplitude R yards, period 2tt q seconds, 
at a constant velocity parallel to the axis and a 
simple harmonic velocity normal to the axis, in the 
plane of the sine curve. Then the root mean square 
prediction error (subject to the same qualifications 
as in the helical case) is RL/ \/2. 

For time of flight 10 sec and for different values of 
the time base c and the period of oscillation, the 
following tables compare the performance of the 
linear and quadratic directors. 

The case q = 6 sec and R = 1,000 ft or more may 
be considered an example of an evasive action. For 
the same time base, the errors cited above for the 
quadratic director are less than those for the linear 


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160 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


director in the case of low frequencies (large q) and 
are greater than those of the linear director for high 
frequencies. As will be seen later, the effects of other 
errors are such that the time base for the quadratic 
director should be two to four times the time base for 
the linear director. For this reason it would appear 
that the quadratic director is not significantly su¬ 
perior to a linear director against the courses con¬ 
sidered above. Tables, of the sort given above, for 
T = 5, 10, 15, and 20 sec are given. 12a 

The numerical results and the formulas for L have an 
import in the case of courses which are far more general 
than the simple helix or the simple sinuosoid. In effect, 
courses of very general types can be approximated 
or built up from simple sinusoids. Suppose that the 
target motion is such that it can be represented in 
the following way. Each rectangular coordinate is a 
sum of sines and cosines in t plus a linear or quadratic 
function of t, where t is clock time. Then the pre¬ 
diction error for the linear or quadratic director, 
respectively, is obtained by combining in quadrature 
the errors of the sort cited above for the separate sine 
and cosine terms: The square of the error on the com¬ 
pound course is the sum of the squares of the errors 
on the component sinusoidal courses. Since very 
general functions can be represented as sums of the 
sort described above c the prediction error that we 
have been considering can be computed for very 
general courses by the above methods. 

As an illustration, consider the target course 

= 250 t 150£ -f- 25 sin -f- 400 cos — , 

2 12 

y = -100 + 10( + 50 sin - , 

6 

z = constant, 

x, y , and z being measured in yards, t in seconds, and 
angles in radians. Consider a time base of 10 sec and 
a time of flight of 10 sec. Let e be the root mean 
square prediction error in yards under the conditions 
of perfect input and no machine error. Then, by the 
above tables, 

2 25 2 (2.8) 2 400 2 (0.71) 2 50 2 (2.4) 2 

e 2 = = 224 2 

for the linear director and 

25 2 (17) 2 4 00 2 (0.29) 2 50 2 (2.1) 2 

e +- ~~ + ~ 2 ' = 319 2 

for the quadratic director. 

c The process of approximating functions by certain sums 
of sines and cosines is considered in works on Fourier series. 


The errors referred to above are the prediction 
errors, supposing that the calculation unit of the 
director has determined the time of flight correctly. 
Actually the time of flight itself is based on the pre¬ 
diction. Accordingly, there will be another contri¬ 
bution to the prediction error due to the fact that the 
time of flight will not be known exactly. 12 

The error function L has the following convenient 
property: 

L n (q,c,T) = L n (kq,kc,kT ), (4) 

where k is any constant different from zero. Approxi¬ 
mate formulas for the function L, together with an 
indication of the range of values in which the formu¬ 
las are valid, are as follows : 


L 2 (q, 


2q*\ 

- + CT + T* 

b « 

& 15, c g 20, 



T g 20; 

1 , 

(& ze-T 

3cT 2 

+ T 3 )’ 

II 

Ol 1 

CO 1 

Uo + 5 + 

2 


y a 15, 

c ^ 

15, T g 15. 


Exact equations for the function L are available. 1215 


10.4.3 Effects of Other Errors 

To be able to make a comparison of the linear and 
quadratic directors, it is proper to consider not only 
the theoretical errors in points of aim that have been 
discussed, but all other errors and random effects 
which enter into the determination of what each shell 
does to the target. An approximate analysis is 
carried out in the following way. 12 There is super¬ 
posed on the calculated theoretical error in point of 
aim a spherically symmetric normal distribution of 
the displacement from the theoretical point of aim 
to the actual point of burst, the standard deviation 
of one component of this distribution being a*. 

The principal elements which contribute to the 
value of a* are the gun and fuze performance errors, 
the machine errors in the mechanism, and the effect 
of the tracking errors. The elements other than the 
effect of the tracking errors are assumed to be the 
same as are applicable to the Army M-9 director and 
90-mm gun. It is assumed that the actual tracking 
errors that would be committed with either director 
are the same as those which have been observed with 
the M-9 director. The effect of the tracking errors in 
the computing mechanism is then estimated by re¬ 
placing the least square directors that we have been 
considering by directors whose extrapolation is of the 


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CURVED FLIGHT DIRECTOR 


161 


following type: The linear director is one which ex¬ 
trapolates along a straight line determined by the 
present position of the target and its position c 
seconds previously. The quadratic director is one 
which extrapolates along the parabola determined by 
the present position of the target, its position c/2 
seconds previously, and its position c seconds previ¬ 
ously. Under these assumptions, the following esti¬ 
mates of the standard deviation <r* are obtained. The 
a* of the quadratic director with time base 20 sec 
is comparable to that of the linear director with 
time base 5 sec. 

Estimated standard deviation a* (in yards) of the spherically 
symmetric normal distribution that is to be superposed 
on the theoretical point of aim. 

Time of flight T = 10 secf 


c 

5 sec 

10 

15 

20 

Linear Director 

161 yd 

106 

89 

82 

Quadratic Director 

1,289 

451 

265 

193 

t Values of <r* also for T = 5, 

15, and 20 sec are given. 

12c 



For purposes of comparison it is assumed 12 that 
the probability of damage is 

_ 1000 | |rV „«) 

P - a * 3 

where r is the distance of the target from the pre¬ 
sumed future position as discussed earlier. The 
probabilities calculated by this formula are not ab¬ 
solute probabilities; but the relative sizes of p* for 
the linear and quadratic directors are presumably 
an indication of the effectiveness of these two di¬ 
rectors. If p* were correct, it would give the propor¬ 
tion of shots which would damage the target. 

The probabilities for the case T = 10 sec are cited 
below. The cases T — 5, 10, 15, and 20 sec have been 

10 7 times the hypothetical probability of damaging the target. 
(The upper entry applies to the linear, the lower to the quad¬ 
ratic, director.) 

T = 10 sec 


\ c 

\ seconds 

Q \ 

seconds \ 

5 

10 

15 

20 

9 

590 

30 

0.02 



5 

100 

290 


12 

1,500 

1,100 

120 

10 


5 

no 

480 

900 

15 

1,500 

3,500 

1,800 

420 


5 

110 

520 

1,200 


considered. 12d Probabilities were calculated only in 
those cases in which the theoretical prediction error 
was less than the radius of the helix or the amplitude 
of the sinusoid. The probabilities refer to a target 
flying either a helical course with a radius of 300 yd 
or a sinusoidal course with an amplitude of 424 = 
\/2 X 300 yd. 

In order to allow for various uncertain effects, in¬ 
cluding the effect of the error in time of flight, and 
also in order to test the sensitivity of the results to 
changes in the estimated <7*, the probabilities were 
also computed with o-^s which were, except for 
rounding, 1.5 times those cited above. The results, 
in the case T = 10 sec, are as follows: 

10 7 times the hypothetical probability of damaging the tar¬ 
get, based on standard deviations 1.5 times those cited above. 
(The upper entry applies to the linear, the lower to the 
quadratic, director.) 


T = 10 sec 


\ c 

\ seconds 

seconds \ 

5 

10 

15 

20 

9 

360 

170 

33 



1.4 

28 

120 


12 

520 

1,000 

500 

43 


1.4 

29 

150 

310 

15 

590 

1,800 

1,400 

1,000 


1.5 

29 

150 

350 


The probabilities are quite sensitive to the standard 
deviation. Nonetheless they indicate that the linear 
director with a comparatively short time base is the 
appropriate fire control instrument to cope with the 
type of maneuver considered. The performances of 
the quadratic director with time base 20 sec is com¬ 
parable to that of the linear director with time base 
5 sec. 

10.4.4 Theoretical Performance of 
Directors against Recorded Course 

The analysis that has been described has been de¬ 
voted to helical and sinusoidal courses and courses by 
implication which might be built up therefrom. It 
seemed desirable to study the behavior of linear and 
quadratic directors on a standard course which might 
be deemed typical of a course flown by an actual 
enemy aircraft. Such a course was supplied by 
Lt. Col. A. H. Musson of the Inspection Board, 
United Kingdom and Canada. It is a two-minute 
course of an enemy aircraft over England. The course 


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162 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


is far from straight, but it perhaps does not represent 
deliberate evasive action. If one introduces rectangu¬ 
lar coordinates in a certain way, the 2 axis being 
vertical, the course can be approximated by the 
following equations: 

x = -0.00001287* 4 + 0.003495* 3 + 0.02644* 2 - 
116.61 - 137.9 

y = —0.00003448* 4 + 0.02367* 3 - 0.02299* 2 - 
67.83* - 4,483 

2 = — 0.00003103* 4 - 0.001670* 3 + 0.16095* 2 - 
5.235* + 1333 

These equations apply to the interval — 60 ^ * ^ 60 
sec; x , y, and z are measured in yards. The x and z 
coordinates have approximately linear trends. The 
y coordinate consists approximately of an oscillation 
of period of the order of 120 sec and of amplitude of 
the order of 1,500 yd. (This can be seen from the 
graphs following page 82 of reference 12.) 

Theoretical errors and probabilities for this re¬ 
corded course, for T = 5, 10, and 15 sec are pre¬ 
sented. 126 The results for T = 10 sec are as follows. 


Errors and probabilities in the case in which the target flies the 
recorded course (described in the text). 

T = 10 sec 


Root mean square error in prediction in j^ards. (Subject to 
the conditions of perfect input, perfect mechanization and 
perfect determination of time of flight.) 


c 

5 sec 

10 

15 

20 

Linear director 

385 yd 

549 

740 

963 

Quadratic director 

47.3 

79 

121 

174 

10 7 times the hypothetical probability of damaging the target. 

(The upper entries apply 

to the linear, the lower 

to the 

quadratic director.) 



c 

5 sec 

10 

15 

20 

Standard deviation a* 

140 

0.02 

0 

0 

as cited above 

5 

110 

480 

930 

Standard deviation 1.5 

210 

3.5 

0 

0 

times those cited above 

1.4 

29 

150 

310 


For the test course, the quadratic director with 
time base 20 sec is the best of those considered. That 
the results obtained for the test course are consistent 
with those earlier obtained may be seen as follows: 
The test course has the general character of a sine 
curve of period 27rl8 sec, that is, with q = 18 sec, and 
amplitude 1,500 yd. According to the earlier theory 
then, the root mean square prediction error is 
1,500L/ n/ 2^ where L = L n (q,c,T) is evaluated at 
q = 18, n = 1 or 2 for the linear or quadratic di¬ 
rector, respectively, c = 20 sec, say, and T = 10 sec. 


Now by the property of the function L given in 
equation (4) 

1,(18,20,10) = L n (9,l0,5); 

the latter quantity has been tabulated. 12a Its value 
is 0.54 for the linear case, and 0.19 for the quadratic. 
This gives errors of 574 and 202 yd, respectively, for 
the linear and quadratic directors; these are of the 
same order of magnitude as the 963 and 174 yd cited 
above for c = 20 sec. 

10.5 ESTIMATES OF CONTRIBUTING 
ERRORS 6 

The estimates to be given apply primarily to the 
Army M9 director and 90-mm gun with preset fuze, 
as of 1945. The estimates are of uneven reliability. 

1. Errors in Static Alignment of Director and Gun. 
It is probable that these errors are very small. 

2. Tracking Errors in Angles. The probable error 
of each component of optical tracking is estimated 
at 0.3 mil for a stable platform; it is also stated that 
this error is roughly proportional to the angular ve¬ 
locity of the target. The probable error of radar 
tracking is 2 mils for the best manually controlled 
equipment. 

3. Tracking in Range. The best radar range has a 
probable error of about 20 yd. The probable error of 
optical range is approximately 

60r 2 

206,2656m yd ’ 

where 6 is the base length in yards, m is the magni¬ 
fication, and r is the range in yards. 

4. Errors in Selsyns Transmitting Data to Directors. 
These are of the order of 0.1 mil and are negligible. 

5. Predicted Position. The errors in predicted posi¬ 
tion depend not only on the errors 1, 2, 3, 4, already 
discussed, but also on the frequencies present in the 
input data (the power spectrum or the input) and 
on the time of flight. For the Army M9 director, 
radar SCR-584, and 90-mm gun, estimates are as 
follows (assuming no evas 

Errors in Predicted Position 
M9 Director; SCR-584 Ra< 

Time of flight 5 

Probable error in angles 
(optical tracking) 3 

Probable error in angles 
(radar tracking) 4 

Probable error along the 
trajectory 

(radar ranging) 40 


ve action): 


20 (sec) 

6 (mils) 
8 (mils) 


75 110 150 (yd) 


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ESTIMATES OF CONTRIBUTING ERRORS 


163 


6. Superelevation , that is, the increment to be 
added to the elevation of the predicted target posi¬ 
tion to compensate for the gravity drop of the pro¬ 
jectile. Errors in superelevation are considered negli¬ 
gible. 

7. Muzzle Velocity. Case I. Preset Fuze. Aspects of 
this question are discussed at length. 3 The muzzle 
velocity varies with the age of the gun, with different 
lots of ammunition, and with different rounds of the 
same lot. Accordingly, the actual muzzle velocity for 
any shot will differ from the input muzzle velocity. 
In some cases, the input muzzle velocity is set in 
accordance with a rule depending in part on the age 
of the gun. A better estimate of the muzzle velocity 
can be attained by the use of a chronograph which 
measures the muzzle velocity of each shot. When the 
chronograph is used, the average muzzle velocity 
of 8 or 12 preceding shots can be used as an estimate 
of the muzzle velocity of the next following shot. It 
is concluded that the probable error in one’s estimate 
of muzzle velocity without a chronograph (using the 
formula MV = 2,730 — 0.087/£ fps, where R is the 
total number of rounds fired by the gun) is 17 fps, 
and with a chronograph is 7 fps in the case of the 
90-mm gun. These errors in muzzle velocity result 
in an error in the burst point of the shell as follows: 

Probable Error in Yards Along the Trajectory , Due to Incorrect 
Muzzle Velocity. (Preset Fuse) 

Time of flight 4 15 25 (sec) 

Error, without chronograph 17 42 66 (yd) 

Error, with chronograph 7 17 26 (yd) 

Case II. Proximity Fuze. In this case, the effect of 
the error in muzzle velocity should be compre¬ 
hended in the gun dispersion (item 13 below; see 
also Section 10.3.2). A numerical appraisal has not 
been given. 

8. Dead Time (Preset Fuze Only). For hand-load¬ 
ing, the probable error in dead time, that is, the time 
between the prediction of the fuze setting and the 
firing of the gun, is about 0.02 sec. This corresponds 
to a probable error along the trajectory of about 
0.06^ yd, in the case of the 90-mm gun, where v is 
the velocity of the target in miles per hour. 9a For a 
target speed of 300 mph, the probable error, due to 
the error in dead time, is thus about 18 yd along the 
trajectory. These figures refer to fuze prediction of 
the conventional type. 

For automatic loading the probable error of dead 
time is of the order of 0.03 sec and has a negligible 
effect. 

9. Errors in Selsyns Transmitting Gun Orders. The 


probable error in the angle transmitted is 0.1 mil. 
The probable error in fuze transmitted is 0.02 sec, 
corresponding to errors along the trajectory of ap¬ 
proximately 13, 7, g,nd 5 yd for times of flight 4, 15, 
and 25 sec, respectively. 

10. Gun Following Errors. The probable error in 
each angle is about 0.4 mil, which is negligible, com¬ 
pared to the errors in the transmitted signal. 

11. Fuze Setting Errors (Preset Fuze). The probable 
error in the case of hand setting is 0.04 sec, corre¬ 
sponding to range errors of approximately 26, 14, and 
10 yd for times of flight 4, 15, and 25 sec, respec¬ 
tively. The probable error in the case of mechanical 
fuze setter is about half that for hand setting. 

12. Fuze Performance Errors (Preset Fuze). For 
times of flight of the order of 20 sec, the probable 
error in fuze performance is from 0.1 to 0.2 sec. The 
resulting probable error along the trajectory is about 
30 yd, more or less constant for different times of 
flight because of the fact that the time fuze error 
increases with increasing time while the residual 
velocity of the shell decreases. 

13. Gun Dispersion. Two types of errors under 
this category have been listed: 6 (1) The errors due 
to faulty estimates in the values of the ballistic wind 
and the ballistic density which are used on the wind 
and density dials of the director. This error is very 
difficult to estimate. Its effect along the trajectory 
has been variously assumed as 10, 20, 40, and 60 yd 
for time of flight of 25 sec. (The latter errors cor¬ 
respond to faulty estimates of range wind, amount¬ 
ing to about 1, 2, 4, and 6 mph, or to errors in air 
density of 0.25, 0.5, 1, and 1.5 per cent. At 15 sec 
time of flight, these errors would be reduced by 
about J^.) The resultant probable error in azimuth 
appears to be of the order of one mil, and in eleva¬ 
tion of the order of 2 mils. 

(2) The error due to the deviation of the ballistic 
characteristics of gun, projectile, and atmosphere 
from those which are presumed in the ballistic 
tables, other than those characteristics discussed 
above (in items 7 or 13-1). The probable error along 
the trajectory from this source is estimated as 10 yd; 
in elevation, as 1 mil; and in azimuth, as negligible. 

Errors in a quantity arising from independent 
sources should be combined in quadrature. Thus, the 
square of the probable error due to causes a and b, 
which are independent, is the sum of the squares of 
the probable errors due to a and to b separately. 

The term gun dispersion has different meanings in 
different problems. For example, the gun dispersion 


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164 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


that was considered in Section 10.2 is not the gun 
dispersion here considered. Dispersion usually refers 
to a variable whose value depends on chance accord¬ 
ing to its probability distribution. In each application 
the variable in question should be precisely defined. 

The reader is referred to a report 3 for a consider¬ 
ation of the variation of muzzle velocity from shot to 
shot, from batch to batch, and with the age of the 
barrel; for a study of errors along the trajectory (part 
of which is subsumed above); and for an actual com¬ 
position of the errors from different sources. 

10.6 PREDICTION CIRCUITS 

The term prediction circuit is here used to denote a 
device which approximates at each instant the future 
value of an input function. The input to the pre¬ 
diction circuit is a quantity: h seconds (the prediction 
interval) and a quantity: x (t) (the predicted function), 
which varies with clock time, t. The output of the 
prediction circuit at the instant t is a quantity z(t), 
which is an approximation of the future value 
x(t + h) of x(t),h seconds later. 

As one application, x may be the fuze setting ap¬ 
propriate at the instant of prediction, in the case of 
heavy antiaircraft with preset fuze, and h may be the 
fuze dead time. Then z will be the fuze setting ap¬ 
propriate at the instant of firing. As another applica¬ 
tion, x may be any coordinate of the target, and h 
may be the time of flight of the projectile. Then z 
will be the coordinate of the target at impact. In the 
latter case, the time of flight h itself must be pre¬ 
dicted by another mechanism or another part of the 
same mechanism. 

Basic requirements of a prediction circuit are that 
its output z{t) approximates the desired future value 
x{t + h) in a satisfactory fashion without undue 
delay and that this approximation is not unduly dis¬ 
turbed when the input function x(t) is itself dis¬ 
turbed. 

10.6.1 Particular Circuits 

A number of prediction circuits have been studied. 9 
Some of the results and methods may be of general 
interest, even though the problem of fuze prediction 
is itself of limited interests 


d The work to be reported was aided by the cordial and 
effective collaboration of the Research Laboratories of the 
Sperry Gyroscope Company. 


The circuits considered are characterized by math¬ 
ematical relations of the form: 

2 = x + y, 

where y is variously defined. 

In the conventional circuit (also called the tan¬ 
gential circuit with feedback), y satisfies the equa¬ 
tion, 

by + y = hx, 

where 6 is a positive constant. Strictly speaking, the 
quantity y is not uniquely determined by the inputs 
x and h, but depends also on certain initial conditions. 
However, the effect of these initial conditions is not 
important after an initial settling time. For example, 
after 4 b sec the difference between any two outputs 
will be less than 2 per cent of their initial difference. 
In effect then, the output of the circuit after settling 
depends only on x and h. The quantity b is known as 
the time constant of the circuit. 

In the Sperry A-circuit, y is an empirical function 
of h and w, where w satisfies the equation 

k\k 2 w + k 2 w w = Xj 

ki and k 2 being positive constants. 

In the Tappert circuit , y satisfies the equation 

by + g(y,h) = hi, (5) 

where g is a function of y and h chosen so as to give 
optimum results. For the particular application to 
the fuze dead time problem, g was determined as 
y — Ci(y + C 2 h) 2 , where C\ and c 2 are constants. If C\ is 
zero, this circuit reduces to the conventional circuit. 

In the tangential circuit with follow-up motor, y is 
the amount of rotation of a motor which receives a 
signal to increase Avhenever y is definitely less than 
hx and to decrease whenever y is definitely greater 
than hx . 

Mechanizations of these circuits are indicated in 
two AMP reports. 9, 10 

Both the Tappert and the Sperry A-circuits are 
interesting refinements of the conventional circuit 
designed to give more effective prediction fairly 
simply. The Sperry A-circuit involves the use of a 
three-dimensional cam. The Tappert circuit involves 
a judiciously chosen and judiciously placed nonlinear 
gear or other nonlinear device. 

The tangential circuit with follow-up motor does 
not seem to provide adequate smoothing of certain 
input disturbances. 10 This result may be of interest, 
as it might have been thought that the lagging effect 
of the follow-up motor would in itself provide smooth¬ 
ing of input disturbances. 


CONFIDENTIAL 




TRIAL FIRE METHODS 


165 


10.6.2 Method of Analysis 

The method of studying the responses of these 
circuits to a perfect input in the case of dead time 
fuze correction circuits was as follows: 9 The func¬ 
tions x were calculated from the ballistic table for 
typical target courses in which the target flew at a 
constant velocity. Because the ballistic functions 
cannot conveniently be represented in explicit ana¬ 
lytic form, the same was true of the functions x. In 
any particular case, one could construct a numerical 
table of x , but no useful explicit analytical expression 
for x in terms of the initial conditions was found. 
Now a numerical description of a family of functions 
is cumbersome at best. In this case, a numerical de¬ 
scription was particularly awkward as the prediction 
circuits made use of x as well as x, and numerical de¬ 
terminations of x were obscured with errors due to 
rounding and errors in reading graphs, and so forth. 
By plotting the functions x, as determined numeri¬ 
cally, against t, it was found that each function x 
could be approximated (according to a least-square 
criterion) by a function of the form 


where a, b, and c were constants depending on the 
course. Thus the family of ideal inputs x(t) was 
approximated by a three-parameter family of ex¬ 
plicit functions for which the difference between the 
output z(t) and the desired output x{t + h) could 
readily be worked out. 

The representation of the functions x as a three- 
parameter family of explicit functions was particu¬ 
larly useful in the determination of the function g of 
the Tappert circuit which makes the circuit most 
effective. The method was as follows: For each course, 
explicit expressions for x and for y (supposing perfect 
prediction) were substituted in equation (5). Thus 
the ideal g for each course was determined as 

g = hx - by. 

A function g of y and h was then chosen which was 
an average of the appropriate functions for the 
different courses. 

10.7 TRIAL FIRE METHODS 

This section applies to projectiles with preset 
fuzes. A trial fire procedure is one which makes use of 


observations of a number of trial shots in order to 
obtain corrections, where necessary, to inputs to the 
director such as muzzle velocity, air density, wind 
velocity, or other factors. Several trial fire procedures 
are described briefly below. Which procedure is most 
satisfactory in a particular set of circumstances de¬ 
pends on those circumstances. It depends also on a 
knowledge of the relative magnitudes of the various 
sources of error to be expected in the firings. These 
points are discussed in detail in a report 1 for the 
Army 90-mm gun. 

If the trial fire procedure is to be based on four 
shots all fired at the same gun settings of the 90-mm 
gun, then no trial fire corrections are warranted if the 
observed center of burst differs from the trial shot 
point, that is, its expected position, by less than 60 
yd in slant range, 2 mils in elation, and 1 mil in 
azimuth. Furthermore, the set up of the gun and its 
equipment should be checked for serious errors if the 
observed center of burst differs from the trial shot 
point by more than 275 yd in slant range, 7 mils in 
elevation or 10 mils in azimuth. 

The trial fire procedures that are considered are the 
following. All involve four trial shots. 

1. Four shots are fired at a single trial shot point. 
The observed deviations of the center of burst of the 
four shots from the trial shot point are corrected by 
spot corrections in elevation, azimuth and percentage 
altitude (or range) settings of the director, a spot 
correction being one which is the same for all tra¬ 
jectories. 

2. Two shots each are fired at two trial shot points 
on the same trajectory. The observed deviations of 
the centers of burst from the trial shot points are 
attributed to muzzle velocity, wind and air density, 
and linear equations are set up to determine the cor¬ 
rections in the latter quantities. 

3. Like item (2), except that the two trial shot 
points are chosen at the same altitude but at different 
horizontal ranges. 

4. Four shots are fired at a single trial shot point 
and the deviation of the center of burst is attributed 
to wind and to a combination of muzzle velocity and 
air density. 

5. Methods of the sort already described, except 
that radar is used to determine range to the points 
of burst. 

6. Two shots each are fired at two trial shot points 
at exactly the same gun settings except that the azi¬ 
muths differ by 90°, thus interchanging the range 
wind and the cross wind. 


CONFIDENTIAL 



166 


STUDIES OF ANTIAIRCRAFT EQUIPMENT 


10.8 TRACER STEREOGRAPHS 

The problem has been considered 8 of measuring the 
amount by which a tracer bullet misses a towed 
target by means of two sets of motion pictures of the 
tracer bullet taken from two different fixed points. 

The Stibitz method consists essentially of finding 
the point of intersection of the apparent tracer paths, 
relative to the target, as given by the two sets of 
motion pictures. The method is an approximation. 
It fails if the tracer curves do not intersect at all, and 
is ambiguous if they intersect more than once. It is 
shown 8 that the method is entirely satisfactory with 
a vertical or nearly vertical base line of about 20 yd, 
and with targets at a slant range of from approxi¬ 
mately 1,000 to 1,500 yd, at elevations of 40° or less, 
provided the residual velocity of the projectile is at 
least about 10 times the velocity of the target. In this 
case, the maximum error in the determination of the 
miss by the Stibitz method is less than about 10 per 
cent of the miss. 

The Radial Grid method does not require a vertical 
or a short base line. It does require, however, either 
quite accurate timing of the different frames (to 
within about sec) or nearly perfect synchroniza¬ 
tion of the two cameras. The operator is supplied 
with a set of radial grids. These are made of trans¬ 
parent material about a foot square and are provided 
with straight lines radiating from a central point, 
marked from 0° to 360°, but not uniformly. From a 
short calculation based on the known approximate 
estimates of the azimuth and elevation of the axis of 
the camera in question, the operator selects the 
suitable radial grid to be used with each of the ap¬ 
parent paths of the tracer, relative to the target. He 
superposes the grid on each apparent path and looks 
for a pair of corresponding readings, one from each 
grid, which contains simultaneous tracers. For ac¬ 
curate work interpolation is necessary. 

10.9 KINEMATIC MODELS FOR 

TRAINING PURPOSES 

There is given 7 a study of a device in which a 
model airplane is exhibited at various distances and 


at various orientations to simulate the appearance of 
an actual airplane in flight, the observer viewing, not 
the model airplane itself, but rather the image of the 
model in a mirror. The device was designed for mili¬ 
tary training purposes. The airplane model is 
mounted on a universal joint on a vertical axis, and 
its orientation is at the disposal of the operator 
through the universal joint. The apparent distance 
of the model from the observer is changed by moving 
the mirror in which the observer views the image of 
the model. 

Formulas and tables are supplied in the report re¬ 
ferred to 7 which solve the problem of determining 
the positions of the mirror and those angles of rota¬ 
tion of the elements of the universal joint which 
would cause the model to produce a simulation of an 
airplane in a specified position or sequence of posi¬ 
tions. 

10.10 NUMERICAL DIFFERENTIATION 
AND SMOOTHING 

In many investigations it is desirable to determine 
the rate of change of a function that is given numeri¬ 
cally at successive instants. As the given numerical 
values are subject to error (due to rounding and per¬ 
haps other causes) and as such errors can have a 
large effect in the calculated rate of change, some 
method of smoothing is appropriate. A method is 
proposed 32 which has been used at Brown University, 
the Frankford Arsenal, and elsewhere. The method is 
as follows: A quadratic function is fitted by least 
squares to each successive 9 values of the function to 
be differentiated, and the first and second deriva¬ 
tives of this quadratic function at the central point 
are taken as the first and second derivatives of the 
tabulated function. The fitting and differentiation is 
carried out, once and for all literally, and formulas 
are obtained for the estimates of the first and second 
derivatives of a tabulated function in terms of the 9 
ordinates or their differences. These formulas provide 
a method for simultaneously differentiating and 
smoothing a tabulated function. Functions of degree 
other than the second and numbers of tabulated 
values different from 9 can be used. 


CONFIDENTIAL 



I 


Chapter 11 


THE RISK TO AIRCRAFT FROM 


11.1 INTRODUCTION 

his chapter is mainly concerned with some 
mathematical problems arising in attempts to 
estimate the probability of damage to an aircraft or 
group of aircraft from one or many shots from heavy 
antiaircraft [AA] guns. Related problems arise in 
air-to-air bombing and in air-to-air or ground-to-air 
rocket fire, but the major part of the mathematical 
analysis so far performed has been devoted to prob¬ 
lems of flak risk. 

The following basic numerical problem underlies 
the whole discussion of flak risk: 

Problem (A). Find the probability of damage to 
an aircraft from a single shot aimed at an arbitrary 
point of space. 

The solution of this problem, once it is attained, is 
used to investigate two major problems of flak risk. 

Problem (B). To estimate the effects of range, 
altitude, speed, arrangement, and spacing of aircraft 
in a given bombing unit on the probability of damage 
to aircraft of the unit from one or many shots. 

Problem ( C ). To study the influence of altitude, 

© 

size of bombing units, and spacing between units on 
the probability of damage to aircraft in a large bomb¬ 
ing mission composed of many bombing units attack¬ 
ing the same gun-defended area. 

The major difference between problems (B) and 
(C) is one of scale; in problem (B) all the aircraft 
under consideration are flying close together in one 
rigid formation; in problem (C) all the aircraft attack¬ 
ing a given target area on a single bombing mission 
are considered. 

Most of the work on problem (A) has been devoted 
to the following special case: 

Problem (D ). Find the probability of damage to an 
aircraft from a single shot aimed at the aircraft itself. 

This special case is of greatest interest in compar¬ 
ing the relative effectiveness of two different shells 
and is, therefore, the question of first interest to 


HIGH-EXPLOSIVE PROJECTILES 

anyone planning A A equipment. A solution of prob¬ 
lem (D) is also useful to an air force since it can be 
used to study risks of damage to a single aircraft in 
terms of slant range and altitude. In addition, an air 
force also needs information about problem (A) in 
order to work out reliable solutions of problems (B) 
and (C) for any formation with more than a single 
aircraft in it. 

This chapter is divided into sections as follows: 

Section 11.2 discusses the intricate process of 
shooting down a bomber by a fragmenting projectile. 

Section 11.3 outlines the mathematical problem in¬ 
volved in the computation of risks from AA fire. 

Section 11.4 is a summary of the basic reports in 
the field including not only work done for the AMP 
but also some earlier reports dealing with problems 
(A) and (D). The first serious work on problem (D) 
available to the writer, and the first considerations 
involving problem (A) are to be found in two British 
reports 14,15 issued by the Exterior Ballistics Depart¬ 
ment in 1940 and 1941. The first of these 14 is mainly 
concerned with developing the method for com¬ 
parison of different shell types; the second 15 is prin¬ 
cipally devoted to a rapid method of finding an 
approximate solution of problem (D) from param¬ 
eters describing the distribution of shell bursts. 
In 1942, in response to a request from the Chairman 
of NDRC, a comparison was made 16 of the probable 
effectiveness of time and proximity fuzes in A A fire 
against high level bombers attacking a ship at sea. 
Soon thereafter work was begun on a related study of 
the dependence of effectiveness of time and proximity 
fuzes on the fragmentation characteristics of the 
shell. The report on this study 13 was distributed in 
1945. The last study of this group 11 uses an approxi¬ 
mate method of solution of problems (A) and (D) in 
order to have some information with which to attack 
problems (B) and (C). 

A similar problem in which the projectile is an air¬ 
borne rocket is described in Section 11.4.5. Many 



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167 


168 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


smaller studies, mostly concerned with comparison 
of effectiveness of different guns and projectiles, are 
grouped together in Section 11.4.6. 

Section 11.5 outlines some of the applications of 
these damage calculations to problems (B) and (C); 
these applications are discussed more fully in the 
flak analysis manual 18 and in an AMP report. 11 

Section 11.6 is a brief concluding remark. 

It is the aim of the chapter to describe a method 
for treating problems of risk. Specific numerical con¬ 
clusions are likely to become obsolete before further 
need for them arises, but the technique by which the 
results were obtained will be useful as long as weapons 
which destroy by means of flying fragments are in 
use. 

It may be well to try to describe, roughly, what 
AMP has and has not contributed to this subject of 
risk from high-explosive [HE] projectiles. The AMP 
does not supply the original experimental informa¬ 
tion on which the computations are based; this comes 
from a variety of sources, mostly Army and Navy, 
OSRD, and British reports. Much of the funda¬ 
mental mathematical theory is contained in any 
elementary statistics text. The theory outlined in 
Section 11.3 is essentially that given in the early 
British report. 14 The principal problems are those of 
obtaining accurate original information (for which 
the AMP could take only what existed), of develop¬ 
ing computational procedures which could be carried 
through before the project became obsolete, and of 
applying these techniques to selected examples. The 
selection of pertinent examples, the development of 
computing techniques, and the carrying through of 
the resulting involved numerical work have been the 
principal contributions of the AMP to the problem 
of risk from flak. The greater part of the work done 
for AMP in the field was carried out by the Statis¬ 
tical Research Group of Columbia University; one 
major report 11 came from the Applied Mathematics 
Group of Brown University and another 13 was 
worked out directly by the AMP staff rather than by 
its contractors. 

11.2 THE PHYSICAL PROCESS OF 
ANTIAIRCRAFT FIRE 

The physical process whose mathematical proba¬ 
bility is to be computed consists of the following 
steps: 

1. The original projectile is launched, arrives at 
some point in space, and bursts. 


2. Fragments fly off from the point of burst in 
many directions at high speed. 

3. Some (or none) of these fragments strike the 
target. 

4. Some (or none) of the fragments striking the 
target do some damage. 

To compute the resulting probability of damage we 
must be able to follow all these steps and collect the 
total risk from all the possible occurrences. Let us 
begin with a discussion of step (1). 

The projectile we are discussing does not arrive in 
space of its own volition but is sent there by some 
mechanical device. If the device were perfect, the 
projectile would always burst in contact with its tar¬ 
get and the probability of damage would be unity. 
However, guns, bombsights, and other existing de¬ 
vices have inaccuracies in design, construction, and 
operation, so the projectile is not very likely to burst 
just where it would be most effective. We must, then, 
consider in each small volume of space the likelihood, 
large or small, that the projectile aimed at the given 
target may burst in that region. 

A great deal of machinery must be used to get the 
projectile to the point of burst. There must be some 
sort of fire control device to estimate the location of 
the best place for the projectile to burst; the pro¬ 
jectile must be fired or launched with the proper 
direction and speed to get it somewhere near the de¬ 
sired point, and there must be some sort of fuze to 
set off the explosive charge in the projectile. 

Let us discuss the problem of the heavy AA gun 
in some detail to get some idea of the imperfections 
that plague the first step in the process of shooting 
down aircraft with ground weapons. The principal 
parts of the mechanism and the function of each part 
is given in the following list. 

1. Tracking instruments, optical or radar, meas¬ 
ure the present bearing, elevation, and range of the 
target and feed this information continuously to the 
fire control computer. 

2. The fire control computer uses the ballistic 
functions of the gun, estimates of the present posi¬ 
tion and rates of the target, and some assumptions 
about the kind of path the target is following to esti¬ 
mate where the gun should be pointed in order that 
a shell fired at present time with the assumed ballistic 
properties will reach the target’s path at the future 
time when the target is there. For time fuzes, the 
computer also estimates the proper fuze setting to 
make the shell explode at the target. 

3. Automatic transmission systems carry this in- 


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THE PHYSICAL PROCESS OF ANTIAIRCRAFT FIRE 


169 


formation to the gun and fuze setter; servomotors 
point the gun continuously in the direction ordered 
by the computer. 

4. The shell is fired from the gun along a trajec¬ 
tory determined by the bearing and elevation set in 
by the servos and by the muzzle velocity with which 
the shell leaves the gun. 

5. If the shell has a time fuze, this explodes the 
shell some time after it leaves the gun. If the shell 
has a proximity fuze (also called VT fuze), and if the 
fuze operates, it explodes the shell if it comes close 
enough to the target, the required proximity de¬ 
pending on direction and rate of closure. 

6. If the gun happens to be mounted on ship¬ 
board, the roll and pitch of the ship under the gun 
and director must be compensated for in steps (1) to 
(4). 

Errors can occur in all these stages of the process. 
Tracking errors are normally small (of the order of 
several mils). These tracking errors tend to increase 
with large angular accelerations. The fire control 
computer has such a complex problem to solve that 
it usually is built with a number of simplifying ap¬ 
proximations so that considerable errors can arise. 
The largest of these (common to all predictors) come 
from two sources: (a) linear prediction for curved 
paths, and (b) the time lags caused by the necessity 
of smoothing the rough information supplied by the 
tracking mechanism and by the time of flight from 
gun to target. The transmission and servo systems 
have a time lag and also have internal oscillations or 
“jitter.” The fire control computer is usually exceed¬ 
ingly complex so that there is a certain amount of 
lost motion in gear trains and servos inside the 
computer itself. 

After the shell is finally fired, it starts out along a 
trajectory which can no longer be influenced by any 
action of the gun crew. The purpose of most of the 
complicated fire control machinery is to get it started 
along a trajectory which will bring the shell close to 
the target. 

If the shell is equipped with a proximity fuze, this 
is all that can be done; the fuze explodes the shell if 
the shell passes near enough to the target under 
suitable conditions of closure. If the shell has a time 
fuze other sources of error are still present. The com¬ 
puter has estimated where the target will be when 
the shell can reach it, has tried to set gun bearing and 
elevation to start the shell trajectory through that 
point, and has tried to set the fuze to explode the 
shell when it passes through that point. In addition 


to inaccuracy of prediction, errors along the trajec¬ 
tory are increased by the variability of time fuzes. 

Once the fuze operates ar(d the shell bursts, the 
next step in the process begins. In this step the nature 
of the shell and it? remaining velocity are the im¬ 
portant factors. The bursting explosive charge shat¬ 
ters the shell case into a number of fragments of various 
weights which fly off with various initial velocities. 
The number, the weight distribution, and the pattern 
in which they fly off from the burst vary with such 
quantities as the thickness of a shell casing, the total 
weight of the shell, the explosive charge, and other 
characteristics of the shell itself. The initial velocity 
a fragment possesses is the vector sum of the velocity 
it would have from the burst of a shell at rest in space 
plus the remaining velocity of the shell at the time 
of burst. Air resistance slows the fragments as they 
fly from the burst until they are going too slowly to 
do any significant damage; this air resistance varies 
with the air density, that is, with the altitude. Also, 
the fragments spread farther apart as they fly from 
the burst; the number of deadly fragments passing 
through each square foot of the sphere of radius 100 
ft about the burst would be only one-fourth of the 
number of deadly fragments passing through each 
square foot of the sphere of radius 50 ft about the 
burst even if there were no air resistance whatever. 

If the burst occurs close enough to the target, 
some fragments may strike with more or less effect. 
If the target is an exceedingly complex device, like a 
heavy bomber, there is still the difficult problem of 
estimating the probable effect of those fragments 
which strike even after we have an estimate of the 
probable mass and velocity distributions of the shell 
fragments at various distances and directions from 
the shell burst. Parts of the target are more vulnera¬ 
ble than others, some parts are more vulnerable from 
one direction than from another, some parts shield 
others. Nothing precise is known about the kind of 
fragment strikes required to kill or injure personnel. 
It is not known what characteristic of the flying 
fragment makes it deadly. Is it the kinetic energy, 
the momentum, or some other combination of mass 
and velocity? This information is needed in deciding 
which fragments to consider at any distance from the 
shell. 

If at this point in the discussion it appears to the 
reader quite improbable that any aircraft is ever 
brought down by AA fire, he will have the correct 
impression as far as single shots at high altitude 
bombers are concerned. Records of American, British, 


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170 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


and German ground-based 90-mm to 5-in. guns col¬ 
lected at various times during World War II show 
that 1,500 to 2,500 shells were fired for each aircraft 
knocked down. This means that single AA guns are 
not much more than a nuisance to attacking bomb¬ 
ers; the Germans made their A A defenses extremely 
dangerous hut only by massed firepower with good 
overall command. 

11.3 THE MATHEMATICAL ANALOGUE 
OF THE FIRE CONTROL PROBLEM 

Because of the extreme complexity of the fire con¬ 
trol problem, which can be solved only approximately 
by any mechanical device, it is not possible to decide 
whether or not a shot fired at a given instant will 
bring down the target; even a complete knowledge 
of the path of the target up to the time the shot was 
fired would not suffice for perfect prediction since the 
target has some considerable amount of time to get 
off its original course after the shot has been fired. 
It would, perhaps, be natural to try to formulate the 
problem of risk from a single shot by asking where 
the shell will burst and whether a burst there will 
destroy the target. Unfortunately these questions 
cannot be answered; we replace them by other ques¬ 
tions phrased in terms of probabilities. 

If a region of space is given, what is the probability 
that a shell will burst in that region? 

If the shell bursts at a given point in space, what 
is the probability that the target will be destroyed? 

How can the total probability that the target will 
be destroyed be computed from the answers to the 
preceding questions? 

These first two questions do not have the same 
answers under all possible tactical situations, for ex¬ 
ample, the answer to the first question certainly de¬ 
pends, to some extent, on the range, altitude, speed, 
and direction of the target, and the answer to the 
second depends greatly on the structure of shell and 
target. However, once numerical expressions are ob¬ 
tained for the first two questions in a given situation, 
the third question can be answered by the compu¬ 
tation of certain definite integrals. 

In Sections 11.3.1 to 11.3.3 we discuss these three 
questions separately. Before going on to these sepa¬ 
rate questions we shall describe certain coordinate 
systems which will be used consistently through all 
that follows in order to help describe the tactical 
situation and the functions to be used in the formula¬ 
tion and solution of the problem. To see why such a 


diversity of coordinates appears in the problem, it is 
only necessary to note that the mere description of a 
tactical situation in which the probabilities are to be 
computed requires knowledge of the (future) range, 
elevation, bearing, direction and speed of motion of 
the target, and the type of gun, shell, and fuze. 

For each shot from a heavy AA gun in a given 
tactical situation, the shell is intended to burst at 
some point O in space at a time when the target is 
at or near that point. We shall set up a rectangular 
coordinate system for locating O from the gun as 
follows (see Figure 1). The origin of the system is at 
the gun, the H axis is vertical, the X axis is hori¬ 
zontal and points in the direction of flight of the 
target aircraft; the Y axis is also horizontal and at 
90° counterclockwise from the x axis, when viewed 
from above. Then the intended point of burst, O, can 
be described by its coordinates (X,Y,H) in this 
system. Various other coordinates are also used 
to locate O. Amo ng these are the slant range D = 
\/x 2 + Y 2 -f H 2 , the ground range R = VX 2 -f- F 2 , 
the angle 6 between the X axis and horizontal pro¬ 
jection of the line from the gun to O (this angle is 
the azimuth or bearing of O from the gun relative to 
the X direction), and E, the elevation angle of the 
line from the gun to O. 

Parallel to the X,Y,H coordinate system take an 
X 0 ,Y 0 ,H 0 system with origin at O. This coordinate 
system is useful in describing the position relative 
to O of target planes at the time of burst. 

We shall ignore the drift of the projectile and as¬ 
sume that the vertical plane through O and the gun 
contains the entire trajectory of the shell. We define 
an x,y,z coordinate system with its origin at the 
point of aim 0, with the z axis pointing away from 
the gun along the trajectory, with the x axis pointing 
upward, perpendicular to the z axis in the vertical 
plane through the trajectory, and with the y axis 
chosen to make x,y,z a right-handed rectangular co¬ 
ordinate system, that is, at 90° counterclockwise 
from the x axis when viewed from the z direction. 
We shall also have occasion to use an a,h,c coordi¬ 
nate system superposed on the x,y,z system when dis¬ 
cussing formations of aircraft in Section 11.5. 

We shall assume, in the neighborhood of O to be 
considered, that the curvatures of the shell trajec¬ 
tories are small enough to neglect, and that these tra¬ 
jectories can be considered as straight lines parallel to 
the z axis. We shall also neglect any yawing of the 
shell and assume that the shell at the time of burst is 
spinning about an axis parallel to the z axis. 


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THE MATHEMATICAL ANALOGUE OF THE FIRE CONTROL PROBLEM 


171 



In discussing the scattering of fragments from the 
point of burst it will be most convenient to use 
spherical coordinates (r,<£,^) where (see Figure 2) r is 
the distance from the shell burst, \f/ is the latitude (so 
that i p = 90° defines the direction of motion of the 
shell and \J/ = 0° defines the equatorial plane of the 
shell), and <£ is the remaining angular variable meas¬ 
ured from the x direction toward the y direction in 
the equatorial plane of the shell. 

Sections 11.3.1 to 11.3.3 discuss the separate ques¬ 
tions raised at the beginning of this section. Together 
they give the mathematical background for solution 
of the problem of risk from a single shot and point 
out some of the difficulties in both theory and appli¬ 


cation. The mathematical technique is essentially 
the same throughout most of the major reports to be 
discussed in Section 11.4; it is that technique which 
is to be presented in this part of the chapter. The 
principal differences in method come in the use of 
special devices and methods of computation rather 
than in the basic theory. 

In one sense it may be said that the present theory 
has yielded about all the general information which 
can be derived from it. To improve the results greatly 
will require new hypotheses in closer relation to the 
physical process and more accurate values of the 
numerical information which must be determined 
experimentally. At present, the effect of the speed of 


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172 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


the target is neglected in most parts of the compu¬ 
tation and is included only in the study of the num¬ 
ber of shots which can be fired while the target is 


x' 



Figure 2. Coordinates of actual point of burst 
(parallel to the x,y,z system of Figure 1 at intended 
point of burst). 


within range (see Section 11.5.1). The magnitude of 
the errors resulting from this neglect of the influence 
of speed on accuracy of fire and on the deadliness of 
the individual fragments is not known. 

n.3.1 The Distribution of Shell 
Bursts 

If in a given tactical situation (as nearly as it 
could be reproduced) a large number of shots were to 
be fired, they would not all burst in the same place 
but would be scattered through a large volume of 
space; the better the fire control mechanism, the 
nearer the center of this scattering of bursts will be 
to the point of aim. 

We assume for each tactical situation that there 
exists a function F(x,y,z) which we shall call the prob¬ 
able density of bursts , such that if a large number N 
of shots were fired in the given tactical situation, 
then the number of shots expected to burst in a cubic 
foot of space around ( x,y,z ) would be N • F(x,y,z). 
Regarded in another way, F(x,y,z) is the probability 
that if a single shot is fired in the given tactical 
situation, the burst will occur in a cubic foot of space 
around ( x,y,z ). By the usual process of setting up a 
definite triple integral (see any good calculus book) 
it can then be seen that if U is any region in space, 
the probability that a single shot fired in the given 


tactical situation will burst somewhere in the region 
U is given by / J fF(x,y,z)dxdydz where the triple in¬ 
tegral is taken over the region U. 

The actual nature of this function F(x,y,z), which 
expresses the probable density of shell bursts through¬ 
out space, is not very well known. Since the integral 



00 

F(x,y,z)dxdydz 

- CO 


expresses the probability that the shell will burst 
somewhere in space, this means that if all fuzes 
operate so that every shell fired bursts somewhere, 
then 



F(x,y,z)dxdydz = 1. 


( 1 ) 


If k per cent of the fuzes fail to operate, the triple 
integral would have the value 1 — k/ 100. However, 
since this is a constant factor for a given fuze type it 
is easier to make the general computations under the 
assumption that all fuzes operate, and then correct 
the last stage of the calculations of risk by multiply¬ 
ing by this operability factor. 

Recalling that the z axis points along the trajec¬ 
tory through the point of aim, we see that the dis¬ 
tribution of bursts normal to the z axis is determined 
by errors and uncertainties in the fire control process 
and by the ballistic dispersion of the gun; the errors 
along the trajectory depend on these and on the fuze 
type as well. Hence the rest of this section falls 
naturally into two parts according as time or proxim¬ 
ity fuzes are used. 


Time Fuzes 

For time fuzes it is generally assumed that the 
errors of the x-, y-, and 2 -coordinate directions are 
independent of one another. Mathematically this 
assumption of independence is formulated by saying 
that F(x,y,z ) is a product of three functions each 
depending on only one of the variables, that is, 
F{x,y,z) = fi{x) f 2 (y) fz{z). Of course the accuracy of 
this assumption is open to question, for example, 
with a fire control predictor using a preset altitude 
reading there is almost certain to be correlation be¬ 
tween errors in the y and z directions (among 
others). 

Since no further information is available about the 
structure of F(x,y,z), it is further assumed that these 
errors in the three coordinate directions are Gaus¬ 
sian distributions with mean zero, that is, it is as¬ 
sumed that the errors in the three coordinate direc¬ 
tions fit the normal error curve for the proper choice 


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THE MATHEMATICAL ANALOGUE OF THE FIRE CONTROL PROBLEM 


173 


of the standard deviation. Such a function has the 
form 


/(*) = 


e -xV2a* 
■%/ 2ttg 


) 


so, multiplying together three such functions with 
the appropriate variables, and using the rules for 
multiplying exponentials, we see that, under all these 
assumptions, the probable density of bursts can be 
expressed in the form 


F(x,y,z) = 


_ g— j W/ffl 5 + 2/V»2 2 + 2 2 / ff 3 J ) 

(V2i) 3 (7 1(7 2 (73 


( 2 ) 


where e is the base of natural logarithms and ai, <r 2 , 
and (7 3 are the standard deviations of the errors in the 
x , y , and z directions, respectively. The values of a 1} 
(7 2 , and (7 3 must be found by experiment or by com¬ 
putations depending on the nature of the whole fire 
control situation. Their dependence on the tactical 
situation is not known at the time this is written al¬ 
though various estimates have been made. 

A distribution law of the form (2) is sometimes 
called an ellipsoidal Gaussian distribution. This name 
is motivated by the shape of the surfaces on which 
F(x,y,z) has any given constant value. If 
e ~l + y s M 2 + « 2 /W) = constant> 

then the exponent is also constant, that is 


x 2 y 2 z 2 

~2 + ~~ 2 + ~2 ~ constant. 

(7i (7 2 (7 3 


It is shown in any elementary analytic geometry 
book that for any positive values of the constant 
this equation defines an ellipsoid with its principal 
axes along the coordinate axes. 

Most of the reports summarized in Section 11.4 
assume that the probable density of bursts is given 
by an ellipsoidal Gaussian law of the form (2). Most 
of these reports make the further assumption that 
lateral and vertical standard deviations are equal, 
that is, that oi = a 2 . Then the probable density of 
bursts is given by a spheroidal Gaussian distribution 
of the form 


F{x,y,z) 


1 2-lW+y*/*?) + (z*/a 3 2 )] 

(\/27r) 3 (7i(73 


0 ) 


Letting 0 * 1 /cr 3 = r this can also be written as 


F(x,y,z) = 


T 

) 3 (7? 


e ~ll (x*+y 2 +T*z*)M 


One report, 15 discussed in Section 11.4.1, uses a 


more general ellipsoidal Gaussian distribution than 
that of equation (2). The formula is 

F(x,y,z) = Kd~ mx ' y ' z) (4) 

where Q(x,y,z) is a general quadratic function of x, y , 
and z which can be written in the form 

Q(x,y,z) = a(x - a) 2 + h(y - /3 ) 2 + c{z - y) 2 
+ 2 'f(y - P)(z - 7) + 2 g(z - y)(x - a) 

+ 2 h(x - a)(y - 0), (5) 

and where K is so chosen that the integral of F(x,y,z) 
over the whole space is equal to one. It is shown in 
any solid analytic geometry text that the surfaces 
Q(x,y,z) = constant are ellipsoids centered at (a, 0 , 7 ) 
when the coefficients satisfy certain conditions. 
These ellipsoids need not, however, be set with their 
axes' along the coordinate axes. This more general 
distribution of bursts is studied in the report . 15 

Another report , 18 which will not be discussed 
further in this chapter, adds in the time errors due 
to the fuze as another Gaussian independent variable 
in addition to the space errors due to prediction and 
then gets the probable density function F(x,y,z ) by 
integrating over the time variable. In this way some 
account is taken of the effect of the speed of the air¬ 
craft on the distribution of bursts. The numerical 
calculations have not been reported, so the magni¬ 
tude of this effect is still unknown. 

Proximity Fuzes 

The errors in the x and y directions are again as¬ 
sumed to be independent. 

A proximity fuze contains a miniature radio 
transmitter and receiver. The fuze, as it travels 
through air, sends out radio waves and receives re¬ 
flected waves from obstacles in the path of the radi¬ 
ation. When the reflected waves satisfy certain con¬ 
ditions, the fuze operates to fire a detonator. The 
point of such operation depends, for a given combi¬ 
nation of fuze and projectile, not only on the size 
and reflecting properties of the target but also on the 
relative orientations and on the relative motion, 
including rate of closure, of projectile and target. 

In the analysis which follows, the target is taken 
to be specified, and its position is regarded as fixed 
in the ( x,y,z) coordinate system. The projectile is 
assumed to travel along a straight trajectory parallel 
to the z axis and is assumed to have a fixed velocity. 
These conditions are approximately fulfilled, for a 
given tactical situation, in a small neighborhood of 
the intended point and time of burst. Along each 


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174 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


trajectory there is assumed to be a theoretical point 
of burst, and the totality of such points is taken to 
be a smooth surface, called the burst surface, defined 
by an equation of the form z = f(x,y). 



travels parallel to z axis). 

For different types of fuze and projectile, the 
general form of the burst surface will vary. How¬ 
ever, it will be assumed for the sake of mathematical 
analysis that this general form is fixed, but that its 
position and size, relative to the target, depend on a 
parameter S which can conveniently be thought of 
as the “sensitivity of the fuze.” It is important to 
note, however, that this terminology is a mere con¬ 
venience. It does not imply any particular prin¬ 
ciple of functioning. It merely lumps together certain 
more or less unpredictable sources of variation in the 
position of the burst surface B s , whose equation, 
for any particular value of S, will be written in the 
form: 

z=fs(x,y). (6) 

Various assumptions will be made as to the de¬ 
pendence of fs(x,y) on the parameter S. These are 
merely working hypotheses to make a suitable 
mathematical analysis possible. For the sake of 
specialized practical applications, modifications 
might be required in the hypotheses and the analysis 
based on them. The principal assumptions and basic 
definitions follow. 

1. Corresponding to each value of S, there is a 
region Us in the (x,y) plane, which is the domain of 
definition of }s{x,y). If a trajectory passes through 
Us, it meets the corresponding burst surface B s and 
the fuze will function. If a trajectory misses Us (and 


hence B s ) a fuze of sensitivity S on that trajectory 
will not function. 

2. If the target is at the point ( a,b,c ), then, for 
each S, fs(x,y) < c. In other words, the burst sur¬ 
face is closer to the gun than is the target, along 
any trajectory. Note that this would not hold for a 
VT fuze with a delay, intended to make it function 
slightly after passing the target. 

3. As the sensitivity S increases, the domain Us 
also increases and the burst surface B s gets farther 
from the target along each trajectory. In otherwords, 
analytically speaking: If Si < $ 2 , then Us 2 contains 
U Sl and fs 2 (x,y) < fsfx,y) wherever both functions 
are defined. 

4. As S varies over its entire range, the burst 
surface B s sweeps out a solid region V in space. 

5. The function fs(x,y) has a continuous partial 
derivative with respect to S. 


x 



Figure 4. The burst volume, the part of space in 
front of the target in which proximity fuzes may 
operate. 

To compute F(x,y,z ) it will also be necessary to 
use the distribution of sensitivities. Suppose that in 
a given tactical situation f(x,y) is the probable 
density of shell trajectories, that is, the probability 
that the shell will pass through a small patch of 
area dA around (x,y) is given by f{x,y)dA. Hence the 
probability that the trajectory will lie in a region A 
of the x,y plane is given by the double integral 

I I f(x,y)dxdy. 


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THE MATHEMATICAL ANALOGUE OF THE FIRE CONTROL PROBLEM 


175 


Similarly, suppose that the probability that the sensi¬ 
tivity lies in the interval from S to S + dS is p(S)dS. 
Then the probability that the sensitivity lies be¬ 
tween Si and S 2 is given by 

J 'Ss 

P (S)dS. 

Si 

We also assume that fuze variations are inde¬ 
pendent of aiming errors. 

Now to compute F(x,y,z) from this we need to 
compute the probability of a burst in some small 
volume dV about ( x f y,z ). If ( x,y,z ) is not in the 
region V, then F(x,y,z) is defined as zero. If ( x,y,z ) 
is in V, then dV will be chosen so small as to lie 
entirely in V, and will be taken as a block bounded 
by a pair of planes x = constant, a pair of planes 
y = constant, and a pair of surfaces S = constant. 
See Figure 5.) The projection of this block on the 

X 


dx 


Figure 5. Typical volume element between two 
burst surfaces. 

x,y plane is a rectangle with edges dx and dy. The 
curved faces of the block are cut off by the surfaces 
B s and B s +ds . We also assume that the fuze sensi¬ 
tivity is independent of the fire control errors. Then 
the probability that the trajectory passes through 
the rectangle is f(x,y)dxdy; the probability that the 



sensitivity lies between S and S + dS is p(S)dS. By 
the assumption of independence of fuze and aiming 
errors, the probability that both these occur is just 
the product 

p(S) dSf(x,y)dxdy. (7) 

Now the volume of the block, since its two curved 
faces are nearly parallel, is approximately dxdydz. 
Since z = fs(x,y), we can hold x and y fixed and 
find that 

dz = j^fs(x,y)dS. 

Hence the volume of the block is 


dV 


dS 


fs(x,y)dxdydS 


and the probable density of bursts around ( x,y,z), 
when z = fs(x,y) is given in terms of x,y, and S by 


F{x,y,z) = 


pjSjdSf^Xjy^dxdy 

dxdydz 


jKSVi^y) 

^fs(x,y) 


( 8 ) 


when z = fs(x,y). Since z = fs(x,y) we can define 
F 0 (x,y,S) to be F[_x } yjs(x,y)~] and rewrite the 
equation as 


F o(x,y,S) 


p(S)f(x,y) 

lkfs(x,y) 


if (x,y) is in U s ; 

F 0 (x,y,S) = 0 (9) 

if not. 

Inside V we can solve equation (6) for S in terms 
of x,y, and z. The solution is unique as a consequence 
of hypotheses (3) and (4), and we write 


S = g(x,y,z) 


if (x,y,z) is in V. Then 

dS = — g(x,y,z)dz 
dz 

and 

p(<S) = vlg(x,y,z)']. 

Substituting in equation (7) and dividing through by 
dxdydz gives 

F(x,y,z) = f{x,y)y\_g{x,y,z)']—^g{x,y,z) 
if ( x,y,z ) is in V; 

F(x,y,z) = 0 (10) 

if (x,y,z) is not in V. 

If we assume (as is usual in fuze calculations) that 
the errors in the x and y direction have independent 


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THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


Gaussian distributions with mean zero, we get 
specialized forms of equations (9) and (10). 


p(S)e~* 


is fs(x ’ y) 


F 0 (x,y,S) = 
if ( x,y ) is in Us) 

F 0 (x,y,S) = 0 
if ( x,y ) is not in Us) and 


— l(x- ! 0 -? +2/ 2 /<r|) 


( 11 ) 


F(x,y,z) = e WM+v ' M) -p\jg{x,y, z)] — g{x,y,z) 

dz 


if (x,y,z) is in V) 

F(x,y,z) = 0 (12) 

if ( x,y,z ) is not in V. 

Most reports that have dealt with damage calcula¬ 
tions for proximity fuzes have made various approxi- 



D 



FLAT TARGET 


Figure 6. Cross sections through burst surfaces. 

mations instead of attempting to carry out this 
involved computation. The first approximation 
assumes that all fuzes have exactly the same sensi¬ 
tivity S so that all shells burst on a single surface Bs. 
As we shall show in Section 11.3.3, this assumption 
reduces the final computation of risk from evalu¬ 
ation of a triple integral to evaluation of a double 
integral. 

Further approximations are made by assuming 
special simple shapes for the burst surface. Two such 
assumptions used in two reference reports 13 * 16 are 


that the burst surface is a hemisphere centered at the 
target with axis of symmetry along the trajectory, or 
that the burst surface is a disk centered on the 
trajectory and lying in a plane at right angles to the 
trajectory. 

The actual shape of the burst surface for a given 
sensitivity S will depend greatly on the size and shape 
of the target. If it is assumed that the antenna of the 
fuze acts like a radiating dipole, the burst surfaces 
for a point target should be symmetric about the 
trajectory through the target and should have cross 
sections resembling the half-circles in Figure 6A. If 
the target is still fairly small and reasonably smooth, 
the burst surfaces will be partly symmetrical with 
cross sections more like those of Figure 6B; if the 
target reflects with entirely different intensities 
from different directions, the burst surfaces will be 
much more lopsided as in Figure 6C. If the target is a 
large plane surface, such as a lake or ocean (targets 
commonly used in testing sensitivity), the burst 
surfaces for a given angle between the trajectory and 
plane surface will be planes parallel to the given 
plane surface. (See Figure 6D.) 

The bulk of the computations that have been done 
with proximity fuzes was carried through before any 
proximity fuzes had been produced, so most compu¬ 
tations carried out by the AMP are carried out for a 
single burst surface of simple type. The reports dis¬ 
cussed in Section 11.4.5 are an exception to this; 
tests with proximity fuzes in airborne rockets were 
available to the men working on that project (AMP 
Study 21). 

Contact Fuzes and Solid Projectiles 

The problem of computing risk from solid pro¬ 
jectiles, such as machine gun bullets, and from ex¬ 
plosive projectiles with contact fuzes is a simplifica¬ 
tion of the problem for proximity fuzes. In this case 
the projectile must pass through some part of the 
target to be effective. The burst surface is then the 
surface of the target turned toward the gun, and the 
distribution of hits is determined by the probable 
density function f(x,y) giving the distribution of 
trajectories cutting the (x,y) plane. 

n.3.2 The Conditional Probability 
of Damage 

The next problem to be dealt with is that of giving 
the risk of damage from a burst which occurs at any 
given point in the neighborhood of the target in the 
tactical situation under discussion. Here we do not 


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THE MATHEMATICAL ANALOGUE OF THE FIRE CONTROL PROBLEM 


177 


consider how the shell comes to burst at that point; 
the problem is: given a point of burst, what is ex¬ 
pected to happen if the shell bursts there? All work 
with this problem so far has required rather strong 
assumptions to be made from a small amount of 
experimental information. 

We can simplify our formulas slightly by assuming 
that the target is at the origin 0. Later we can easily 
account for the trivial effect of moving the target to 
another point. We shall attempt to find a function 
p(x,y,z) to describe the probability that a shell burst¬ 
ing at ( x,y,z ) will destroy the target at the origin. We 
shall call this function the conditional probability of 
damage. 

The values of p(x,y,z) depend on many character¬ 
istics of the shell and target. These factors are the 
weight and shape of the fragments from the bursting 
shell, the speeds and directions with which they are 
thrown off from the burst, the air density, the size, 
shape, and orientation of the target, the arrange¬ 
ment in the target of vulnerable parts such as pilots 
and fuel tanks, the vulnerability of these parts to 
fragments of the kind thrown off from the burst, the 
speed of the target, and its direction relative to the 
shell trajectory. 

It should be noted here that many of these factors 
are ignored or suppressed in most calculations of 
risk. A common broad assumption is that the target 
can be represented by a sphere hanging in space at 
the point of aim; this assumption removes all effect 
of the speed of the target on the conditional proba¬ 
bility and reduces the effects of the complex vulner¬ 
ability characteristics of the target to the choice of a 
single variable, the size of the sphere. 

For either an ideal or an actual target, the first 
step in computing p(x,y,z) is to describe the frag¬ 
mentation pattern of the shell used. This description 
is simplest in the spherical coordinate system r,(t>,\f/ 
shown in Figure 2 where the line \p = 90° points 
along the shell axis toward the nose of the shell. If 
we neglect yaw of the projectile and assume that the 
shell points along the trajectory, this makes the 
line \f/ = 90° parallel to the z axis. 

Because of the spin of the shell and its axial sym¬ 
metry we can assume that the number of fragments 
of a given mass and speed thrown out by the burst 
will be independent of the longitude angle </>. It is 
generally assumed that the effectiveness of a flying 
fragment must depend on its shape and mass, and on 
the relative speed of fragment and target at the 
instant the fragment strikes. 


If the target speed is small enough, it makes no 
great difference to the effectiveness of a fragment if 
the target speed is neglected entirely and it is as¬ 
sumed that the target is stationary in space at the 
instant the shell bursts. For high speed targets the 
effectiveness of a fragment, particularly of a heavy 
fragment that can do damage at speeds of a few 
hundred feet per second, is largely dependent on 
whether it is going in the same or the opposite 
direction as the target. 

Neglecting the target speed greatly simplifies the 
calculations of the conditional probability because 
it simplifies the problem of deciding which fragments 
to count. When target speed is zero, the relative 
speed of a fragment reduces to its speed through the 
air; as with mass and shape, there is no reason to 
believe that this speed varies with the longitude. 
Hence in case target speed is zero, we can choose 
fragments for consideration by criteria involving 
weight and speed and still see that the expected dis¬ 
tribution of these selected fragments is independent 
of the longitude angle </>. 

In most of the work done heretofore, the fragments 
to be counted have been selected on one of two bases: 
first, by putting up screens at various distances from 
a bursting shell and counting the number of frag¬ 
ments that penetrate the screens, and second, by 
assuming that a fragment must have a certain 
minimum kinetic energy to do damage and counting 
only those fragments with at least that much energy. 

We have a choice of several similar methods of 
describing the pattern of fragments from a shell 
burst providing only that it is independent of longi¬ 
tude. 

1. The angular density p{r,\J/) of fragments at any 
point with coordinates (r,0,^) is the number of 
fragments per solid unit angle a passing through the 
sphere of radius r about the burst in a small region 
about the direction denoted by (0,^). 

This angular density p{r,\p) is written without <£ to 
show its dependence on r and \f/ alone. To estimate it 
for a given position ( r,(f>,4 /), count the fragments 
going through a given small area A on the sphere of 
radius r, about any point at latitude »p and divide 
this number of fragments by the solid angle A/r 2 
subtended by the area A. 

2. The area density of fragments is the number of 
deadly fragments per square foot of surface at dis- 

a A unit solid angle is a cone reaching out from the origin 
which cuts off an area of 1 square unit on a sphere of radius 
1 unit. 


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THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


Z' 




Figure 7A. Typical region for estimating angular 
fragment density, z' axis points toward shell nose. 
Figure 7B. Typical zone for calculating zone count. 


tance r and latitude \p. It is found from the formula 
p(r,\f/)/r 2 . 

3. The zone count n(r,\f/) is obtained by dividing 
the sphere of radius r around the burst into a number 
of zones, say of width 2.5°, and counting the number 
of fragments passing through each zone. The term 


n(r,\f/) is the number of fragments in the zone con¬ 
taining \J/ on the sphere of radius r. 

As the fragments fly further from the burst they 
are slowed by air resistance so that fewer of them are 
effective; hence p(r,\p) and n(r,\p) decrease with in¬ 
creasing r. h If the weight, speed, and ballistic shape 
distribution of fragments is independent of \f/ we can 
break p(r,\J/) into a product of two factors 

p(r,t) = p{i)g(r) (13) 

where p(\J/) defines the angular density of fragments 
which have energy enough to do damage very close 
to the burst, and g(r) describes the decrease in the 
number of deadly fragments with distance r from the 
burst. Similarly we could write 

n{r,i) = n(\f/)g(r) (14) 

with the number of fragments in the zone about 
^ with energy enough to do damage very near the 
burst, and g(r) the same function as in equation (13). 
The term g(r) is called the fall-off law for effective¬ 
ness of fragments. 

Various types of experiments have been performed 
to find this angular fragmentation pattern and to get 
estimates for the fall-off law g(r). 

1. Wooden screens ruled off in square feet are 
placed at various distances from the bursting shell 
and the number of fragments penetrating each 
square foot area is counted. This gives values of 
p(r,^) [or n(r,\p )] for several particular values of r 
and some values of \p. However, the values of i p 
must be corrected in some way for the fact that the 
shell in the test is at rest while the shell in any tactical 
situation has some considerable remaining velocity 
which tends to throw the fragments in the moving 
burst pattern forward of the burst zone in which 
they appear in the static burst patterns. 

2. Shells are burst in sand pits and in sandbag 
beehives; the fragments are sifted out of the sand 
and sorted by weight or by the fineness of the screen 
they sift through. These tests are to determine the 
distribution of weights of the fragments of the shell 
and give no information about the angular fragment 
pattern. 

3. Various photographic and penetration tests 
have been devised to measure the speeds at which 
fragments are thrown off from the bursting shell. 
These are usually static tests and give values in the 


b There are some indications that 50 or 100 ft from the 
burst, fragments may still be breaking apart. One experi¬ 
ment 14 shows no decrease in n(r,\J/) as r went from 40 to 60 ft. 


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THE MATHEMATICAL ANALOGUE OF THE FIRE CONTROL PROBLEM 


179 


neighborhood of 2,500 fps for the initial fragment 
velocity. 

4. Firing against actual aircraft on the ground 
under conditions where the location of the burst can 
be determined. This helps to determine the relation 
between angular density of fragments and the con¬ 
ditional probability of damage to aircraft. 

In terms of the angular fragment pattern it is not 
difficult to estimate the expected number of hits on 
an aircraft from its size and distance from the burst. 
However, the aircraft is a complex target and frag¬ 
ment hits on parts of it are only a minor nuisance, 
not fatally damaging. To pass from the fragment 
density p(r,\f/) to the number of hits on a target pre¬ 
senting the area A toward the burst at distance r 
away, we need only multiply the angular fragment 
density p(r,\f/) by the solid angle A/r 2 , subtended by 
the target. If we are interested in the expected num¬ 
ber of damaging hits due to the fragments, we can use 
the same formula provided we use for A not the 
actual area presented by the target, wings, tail sur¬ 
faces, and all, but only that area vulnerable to the 
fragments being used in the computations. This 
concept of the vulnerable area of the target is, of 
course, a fiction but an extremely useful one in 
making computations of the conditional probability 
of damage. Estimates of vulnerable area are usually 
arrived at by comparing experiments of the types (1) 
and (4) just referred to. 

Data of the four types (1) to (4) have been collected 
by a number of organizations. A considerable body of 
data on fragmentation characteristics of U.S. Army 
projectiles is collected in a series of TDBS Reports. 0 
A report of Section T of OSRD 17 contains the most 
exhaustive study extant of conditional probability 
and vulnerable area derived by test firing against 
aircraft on the ground. The British have also made 
a number of studies of fragmentation characteristics 
of their own shells. 

Once the vulnerable area A is found, the expected 
number of damaging hits on the target at (r,<t>,\l/) is 
given by the formula 


m(r,\p) = 


^pQvA) 

y*2 


The next problem is to pass from the expected 
number of damaging hits on the target to the proba¬ 
bility of damage to the target. This depends on how 
nearly the actual scattering of fragments from a 


c Technical Division, Ballistics Section, Office of the Chief 
of Ordnance. 


single burst conforms to the expected pattern for 
shells of its type. It is generally assumed that this 
scattering of fragments follow^ the so-called Poisson 
law (which has been observed in other problems of 
random scattering). This law implies that if k is a 
whole number and m is the expected number of hits 
on a given target, then the probability of getting 
exactly k hits is 


That is, the probability of 0, 1, 2, 3, • • • , hits is 


e~ m , me~ m , 


m 2 e~ m m 3 e~ m 
2! ' 3! 


Hence the probability of getting at least one hit is 1 
minus the probability of getting no hits, that is 


p(x,y,z) = 1 - e ~ m , (15) 


where m = m(r,\f/) and ( r ,0,^) is the location of the 
origin 0 from the point ( x,y,z ) at which the burst 
occurs. Some experimental justification of this equa¬ 
tion is given in a reference report. 14 

If we are interested only in the problem D of the 
introduction of finding the risk to a single aircraft 
at the point of aim, then p(x,y,z) and F(x,y,z) can 
be combined to obtain the probability of damage. 
To solve problem A, however, we must have the 
conditional probability that a shell bursting at 
( x,y,z ) will destroy a target at some point ( a,b,c ). 
This new probability is easily computed from what 
we know already, for moving both burst and target 
by the vector (— a, — b, — c) we see that the risk to a 
target at ( a,b,c ) from a shell burst at ( x,y,z ) is pre¬ 
cisely the risk to a target at (0,0,0) from a shell 
burst at ( x — a,y — b,z — c), that is, the risk is 
equal to p(x — a,y — b,z — c). 

It is to be remembered that in passing from num¬ 
ber of hits to number of effective hits, we tacitly 
assumed that the vulnerable area A did not de¬ 
pend on 9; that includes the assumption that the 
speed of the target and the aspect from which it is 
viewed have no effect on the conditional probability. 
No careful study of this has ever been completed, 
for computational difficulties would be considerable. 
The effect of target speed has been considered, 19 but 
numerical results have never been published. 


n.3.3 The Probability of Damage 
to a Single Aircraft 

Once the probable density of bursts and the con¬ 
ditional probability of damage are given, the calcu- 


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180 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


lation of the risk from one shot at a single aircraft in 
a given tactical situation can be calculated by means 
of a triple integral. Let P(a,b,c) be the probability of 
damage to a target at ( a,b,c ) from one shot fired at 0 
in the given tactical situation. To set up the triple 
integral for P(a,b,c), cut the space around 0 into 
small bricks by planes parallel to the coordinate 
planes, and consider any such brick with center at a 
point ( x,y,z ) and edges of length dx , dy, and dz. 
Then, from the definition of probable density of 
bursts, the probability that the shell will burst in 

X 



Figure 8. Volume element dxdydz. 


this region is F(x,y,z) dxdydz. If the shell bursts 
there, the probability that it will damage the 
target at ( a,b,c ) is p(x — a,y — b,z — c). Hence the 
probability that the shell will burst in this small 
region and damage the target at ( a,b,c ) is the 
product 

p(x — a,y — b,z — c)F(x,y,z) dxdydz. 

Summing up over all such small regions, we see that 
the risk P(a,b,c) can be given by the triple integral 


P(a,b,c ) 


iff 


p(x — a,y — b,z — c)F(x,y,z) dxdydz. 

(16) 


In case the target aircraft is at the point 0, we 

have a = b = c = 0 and the risk for such an aircraft is 

00 

P(0,0,0) = JJJ p(x,y,z)F(x,y,z)dxdydz. (17) 

— 00 

It is this last formula which has been used by most 
writers trying to solve problem (D) for time fuzes; 
the formula for P(a,b,c) is needed in solving problem 
(A) and therefore has applications to problems (B) 
and (C). (Some discussion of this is contained in 
Section 11.5.) 

For proximity fuzes and a single target at the 
origin, the integral has usually been replaced by a 
simpler double integral by making the assumption 
that all fuzes have the same sensitivity, that is, that 
all the shells burst on a single surface in front of the 
target. That surface has an equation of the form 
z = f(x,y). Assuming independent Gaussian errors in 
x and y directions, the probability that the shell will 
pass through a small rectangle of sides dx and dy 
about (x,y) is 1 

2'7TO'i0’2 

If the shell is on the trajectory through ( x,y ), then 
either it is too far away from the target to burst at 
all or it is in the region U where f(x,y) is defined and 
the shell bursts at [x,yj{x,y)~\] hence the probability 
that a shell on the trajectory through ( x,y ) will 
damage the target at 0 is p[_x,yj(x,y)~] if {x,y) is in 
t/, and is zero if not. Hence the risk from trajectories 
passing through the small rectangle is 

Plx,yJ(x,y)y ~ l(3 * /91 * + yi/a ^dxdy 
and the total risk to a target at the origin is given by 

P(0,0,0) = ffA.x,yJ{x,y)le- iW '''+» ,/n ' ) dxdy. (18) 
u 

P(a,b,c ) could be computed by a similar process. 

It is worth while to mention that risk from solid 
projectiles and contact-fuzed explosive projectiles 
can be computed from the formula (18) provided 
that proper interpretation of the various terms is 
made. For contact-fuzed projectiles the analogy is 
extremely close. The burst surface for contact-fuzed 
projectiles is the surface of the target turned toward 
the negative z axis. The term p[_x,y,f{x,y)~\ is a 
function p'(x,y) determined by the risk from a burst 
of the given shell at that point on the surface of the 
aircraft; the risk is then given by equation (18), 
which can also be rewritten 

P(0,0,0) =JJp'(x,y)e- l<x ’ /n ‘ +,/W, dxdy. (19) 

u 


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HISTORICAL SUMMARY 


181 


If p'(x,y) is regarded as the probability that a 
given solid projectile entering the aircraft on the 
trajectory determined by (x,y) will damage the air¬ 
craft, then this same formula can be used for solid 
projectiles. This shows that there is a close analogy 



Figure 9. Area element and burst surface. 


between the risks from contact-fuzed and proximity- 
fuzed projectiles; the difference is largely a differ¬ 
ence of scale. The contact-fuzed shell may be effec¬ 
tive if it passes through the target; the proximity- 
fuzed shell may be effective if it passes through a 
certain larger region near the target. 

We may note that if a computation of P{a,b,c) or 
P(0,0,0) is made from one of these formulas on the 
assumption that all fuzes operate, then, to estimate 
the effect if k per cent of the fuzes fails to operate, 
it is only necessary to put a factor (1 — k/ 100) in 
the probable density of bursts. Since this is a con¬ 
stant factor, it can be removed from under the 
integral sign in these equations for risk so the risk 
is easily corrected for operability of the fuze. 

In applying the various formulas of this section 
it is soon discovered that the integrals can seldom 
be evaluated in any simple form but must be com¬ 
puted numerically. For any given special case, this 
evaluation is not too difficult; however, it becomes 
extremely tedious and expensive if computations 
must be made for a large number of examples. 

The integrals for risk from time and proximity 


fuzes are usually evaluated by converting them to 
the (r,</>,^) coordinate system with origin at the 
shell burst. 

11.4 HISTORICAL SUMMARY OF PRIN¬ 
CIPAL STUDIES OF RISK TO AN 
AIRCRAFT FROM A SINGLE SHOT 

In this section we describe the specific problems 
studied in certain reports on risk of damage from 
high-explosive shells and rockets, outline the as¬ 
sumptions and the nature of the results, and give 
some ideas of the methods used. 

Section 11.4.1 deals with two British reports; 14> 15 
Section 11.4.2 discusses OSRD Report No. 738, 16 the 
first major study of this kind made in this country. 
Section 11.4.3 is devoted to another such study 
started soon after OSRD 738 but published years 
later as AMP Note No. 19. 13 Section 11.4.4 deals 
with the last large study of flak risk, AMP Report 
No. 185.1R. 11 Section 11.4.5 discusses some questions 
of risk to a bomber from airborne rocket fire. 3-5 
Section 11.4.6 discusses some AMP reports of 
special aspects of the flak problem; for instance, one 
such report discusses the probability of shooting 
down a directly approaching aircraft. 8 Other re¬ 
ports in this section deal with comparison of different 
guns and projectiles, usually for directly approach¬ 
ing aircraft. 

The attitudes with which these reports regard 
problem (D) vary widely because of the difference in 
the specific problems that required some study of 
risk. In the first British report, 14 the emphasis is on 
the comparison of the effectiveness of different types 
of AA shells and of different standards of accuracy 
in shooting. A large part of the report is also devoted 
to a full development of the theoretical methods 
used in the study of risk. The second British report 15 
is vitally concerned with the effect of the distribu¬ 
tion of shell bursts on the probability of damage and 
attempts to get a formula for quick approximate 
calculation of the risk when the probable density of 
bursts is given in a certain form. OSRD Report No. 
738 16 is mainly concerned with comparison of the 
effectiveness of time and proximity fuzes for a given 
shell in half a dozen different tactical situations. 
AMP Note No. 19 13 also compares the risk for time 
and proximity fuzes; a tactical situation is fixed and 
the emphasis is placed on the effects of variations in 
the fragmentation characteristics of the shell. AMP 


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182 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


Report 185.1R 11 attempts to solve problems (A), 
(B), and (C) of the introduction (Section 11.1) and 
therefore gives an approximate solution of problems 
(A) and (D) which can be computed for many dif¬ 
ferent tactical situations; in particular, for many 
different ranges and altitudes of the target. 

11.4.1 Two British Studies 

In 1940, the British developed a method 14 for the 
comparison of the effectiveness of different A A shells 
under varying assumptions about the accuracy of 
the AA fire. The mathematical argument is, essen¬ 
tially, that outlined in Section 11.3 of this chapter 
for time-fuzed shells. 

The probable density of bursts is assumed to be 
Gaussian with errors independent in the three co¬ 
ordinate directions and, moreover, it is assumed that 
the standard deviations and a 2 in the x and y 
directions (across the trajectory) are equal. Then the 
probable density of bursts takes the form 

F(x,y,z) - 4 e -lW«W+'W>. 

(V27t) 3 o- 1 o-3 

The fragmentation data came from several sources: 

1. Counts of the number of fragments penetrating 
wooden screens placed at different distances from the 
shell burst. 

2. Counts of the number of disabling hits on a 
Blenheim bomber hung in the main fragment zone 
of the shell bursts. 

3. Counts of number and weight distribution of 
fragments from shells exploded in sandbag beehives. 

From the first source it was possible to estimate 
the fragment density p(r,\f/) in terms of the distance 
and direction of the target from the shell. By com¬ 
paring (1) and (2) a correlation between fragment 
density and damage to a Blenheim bomber was 
found for the type of shell used in this test. From (3) 
information as to the fall-off of the number of deadly 
fragments with distance from the shell is obtained 
for a number of different shells. 

The actual complex target is replaced by a hypo¬ 
thetical target presenting a vulnerable area of 100 
sq ft from any distance or direction. The fragmen¬ 
tation pattern has been described by dividing the 
hemisphere forward of the burst into zones of 2.5° 
width and giving the number of fragments in each 
zone which penetrate the 2-in. wooden screen of test 
(1) at the distance in question. The number of these 
fragments expected to strike a disk of 100 sq ft area 
was computed as follows: If the center of the disk is 


at the disk itself overlaps some of the 2.5° 

zones. In each zone divide the area of the part of the 
disk in that zone by the area of the zone and multiply 
by the number of fragments in that zone to get the 
expected number of hits from that zone. Add the 
results to get the expected number of hits on the 
disk (at the range r and latitude i p). 

The comparison of tests (1) and (2) suggested that 
the Blenheim bomber received one damaging hit for 
each seven “throughs” (fragments penetrating a 
100 sq ft area of wooden screen) at the same latitude 
and distance from the burst. Hence m(r,\p), the ex¬ 
pected number of damaging hits on an aircraft at 
range r and latitude \f/, was found from the screen 
tests and fall-off data. Direct counts of the number 
of “throughs” were not available for values of r 
greater than 150 ft, so the dependence of m{r,yp) on r 
for larger r was estimated from the fall-off of velocity 
of the fragments flying through the air. 

Once m (r,xf/) is given, the probability of at least one 
damaging hit is, from the Poisson law, 1 — e~ m , 
where m = m(r,\f/). An appendix of this report 14 dis¬ 
cusses in some detail the applicability of the Poisson 
law to the scattering of shell fragments; the evidence 
then available fitted very well. 

The integration of the triple integral was done 
numerically in such a way that the portion of the 
risk due to certain regions around the shell could be 
found. It was found, for example, that under many 
circumstances more than half of the risk comes from 
bursts more than 100 ft from the target. 

This early report is fundamental in the field; it 
contains a clear discussion of the aims and difficulties 
in attempting to find the risk from a single shot. 

A second British report 15 describes a method of 
computing the probability of damage when the 
probable density of bursts does not have the simple 
form 

_ 1 3 2 ) 

(\/27r) 3 

given in equation (2) but follows the more general 
ellipsoidal Gaussian law given in equations (4) and 
(5); that is 

where 

Q(x,y,z) = a(x - a) 2 + b(y - fi) 2 + c(z - y) 2 
+ 2 f(y - 0)(z - y) + 2g(z - y)(x - a) 

+ 2 h(x - a) (y - 0). (5) 


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HISTORICAL SUMMARY 


183 


Q is any general quadratic form whose level surfaces 
are ellipsoidal, and K is chosen so that the integral 
of F(x,y,z) over all space is one. Most of this paper 
deals with the case of no bias, that is, a = (3 = y = 0. 

By using certain approximations in the numerical 
evaluation of the triple integral for probability of 
damage, a function J 0 (Ao) is defined in terms of the 
type of shell used, the altitude of the burst, and the 
remaining velocity of the shell. This function is 
tabulated for two shells (British 3.7 in. and 4.5 in.) 
at altitude 10,000 ft and remaining velocity 1,350 
fps. Two approximations to P (0,0,0) are given in 
such a form that they can be computed quickly 
from the coefficients of the quadratic form Q(x,y,z). 
The simplest approximation is 

P(0,0,0) = P7o(Ao), 

where 

a T" b -f- c 


and a, b, and c are taken from the given polynomial 
Q. A somewhat more accurate approximation is 


where 


Ao 


P(0,0,0) = A/o(Ao), 

(~ 2 ~) C ° S2 \ sin2 


and rpo is an angle depending on the direction of the 
main fragment zone of the bursting shell and, hence, 
depending on the type of shell and its remaining 
velocity at the time of burst. These approximations 
are reasonably good as long as the standard devi¬ 
ations are greater than 100 ft. Tables are given for 
correction of these approximations to other altitudes 
and remaining velocities, so a quick estimate of risk 
for either of the two shells considered can be given 
in any tactical situation in which the probable 
density of bursts can be described in the form of 
equations (4) and (5), with a = = 7 = 0. 


n.4.2 OSRD Report No. 738 

This report 16 compares the effectiveness of time 
and proximity fuzes for Navy 5-in., 38-caliber shells. 
Since the report was written before any knowledge 
was available of actual burst surfaces of a proximity 
fuze, computations were carried through for two 
types of surfaces, hemisphere and disk, and for a 
wide variety of sensitivity ratios. 

The probable density of bursts for time fuzes was 
assumed to follow a spheroidal Gaussian law (equa¬ 


tion (3)) with standard deviations o x and o- 3 across 
and along the trajectory. Dependence of the risk on 
o i and (7 3 was studied along with the dependence on 
fuze characteristics. The computations for time fuzes 
are made from the formulas (3) and (17). The com¬ 
putations for proximity fuzes were made from the 
formula (18) with oi = o 2 , with U a circle of radius 
b feet centered at O and with f(x,y) = VV — x 2 — y 2 
for the hemispherical burst surface of radius b, and 
f(x,y) = — d, a constant, for a disk burst surface of 
radius b centered d feet in front of the target. 

Various tables give: 

1. Pi, the risk from a time-fuzed shell, in terms of 
the standard deviations oi and cr 3 across and along 
the trajectory. 

2. P 2 , the risk from a proximity-fuzed shell with 
hemispherical burst surface, in terms of the standard 
deviation o x across the trajectory and of the radius 
of the hemisphere. 

3. P 3 , the risk from a proximity-fuzed shell with 
a disk burst surface, in terms of the standard devi¬ 
ation o\ across the trajectory, the radius b of the 
disk and the distance d from the center of the disk 
to the target. 

4. The advantage ratios P 2 /P 1 and P 3 /Pi in 
terms of these parameters. 

It may be noted that for the range of values of 01 
used in this report, the risks from proximity fuzes 
vary approximately inversely with the square of o h 
the standard deviation across the trajectory. That 
is to be expected, for the proximity fuze risk behaves 
much like the risk from a solid projectile. The differ¬ 
ence is in the area through which a proximity-fuzed 
shell may pass and still be effective. In both cases 
there is a certain area of the x,y plane such that a 
shell on a trajectory passing through a given point 
( x,y ) of that area has a certain probability of doing 
damage. For solid or contact-fuzed projectiles, this 
area is the presented area of the target; for proximity- 
fuzed projectiles, this area is the presented area of 
the burst surface. For large values of <ri, the function 

2tto\ 

is approximately equal to over the burst 

surface. Substituting in equation (18) and taking 
the constant factor 1/(2tto\) from under the inte¬ 
gral sign leaves an integral not dependent on <n; 
hence the risk is approximately proportional to 

l/o?. 

Some study is also made of the effect of altitude 


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184 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 



Figure 10. Optimum fragmentation patterns for various total numbers of fragments (time fuze shell; 5738 fall-off law). 


on these risks. This information is used to find the 
advantage ratio of proximity over time fuzes in half 
a dozen different tactical situations specified by 
altitude, slant range, and fire control errors. 

The results of this study indicate that in the typical 
tactical situations considered proximity fuzes would 
give from 2% to 5 times the probability of damage 
from time fuzes. The range of variation in relative 
effectiveness is due to the variations in aiming errors, 
altitude, and shape of burst surface. In spite of the 
uncertainties in the computation, this report indi¬ 
cated that if proximity fuzes could be made to 
operate even half or three-fourths of the time, then 
proximity fuzes would be measurably more effec¬ 
tive than time fuzes. This prediction has been borne 
out by combat records. The results of the next 
study 13 also emphasize the same conclusion although 
the formal final report was not issued until long after 
proximity fuzes had been in wide use. 

One section of the report compares the variation 
of effectiveness of shells of different weights. Another 
section discusses the dependence of the effectiveness 
on the weight distribution of fragments from the 
burst. This has its effect on the rate at which the 


fragments lose speed as they fly from the burst and 
on the speed a fragment must retain to have enough 
energy to do damage when it strikes the target. 

In both this and the preceding report the target is 
assumed to be divided into three parts of different 
vulnerabilities (these parts are intended to corre¬ 
spond to pilot and bombardier, fuel system, and 
motor). The term m(r,\f/), the number of fragments 
effective against this vulnerable area in direction \p 
and distance r, is computed from two sorts of data; 
first, the angular fragmentation pattern of the shell 
under discussion which gives the number of frag¬ 
ments in each zone at the burst itself, and, second, 
sand pit tests giving the number of fragments of 
given weight thrown off by the burst. Assuming that 
the weight pattern of fragments is the same in all 
zones, the number of fragments still effective at dis¬ 
tance r is computed from the law for retardation of 
fragments by air friction and from the energy needed 
to be effective when the target is struck. 

n.4.3 AMP Note No. 19 

This report, 13 like OSRD 738, does not take in¬ 
to consideration the functioning characteristics of 


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HISTORICAL SUMMARY 


185 


proximity fuzes. Both reports compare time and 
proximity fuzes, but, in the later one, a particular 
tactical situation is specified and a study made of 
the dependence of damage probabilities on the fol¬ 
lowing fragmentation characteristics: (1) angular 
distribution of fragment pattern, (2) total number of 
fragments, (3) the fall-off law for effectiveness, and 
(4) the weight distribution of fragments. The in¬ 
vestigation compares time fuzes, proximity fuzes 
with hemispherical burst surfaces, and proximity 
fuzes with disk burst surfaces. Such burst surfaces 
are recognized as hypothetical, but the methods de¬ 
veloped are of wide applicability. 

For each fuze type, individual angular zone con¬ 
tributions to the total probability of at least one 
damaging fragment hit are analyzed. The zones are 
measured in terms of latitude relative to the equa¬ 
torial plane of the shell. One of the principal results is 
the development of optimum fragmentation patterns 
for each type of fuze and for various given total 
numbers of fragments. In the case of a time fuze, for 
example, the optimum pattern concentrates most 
of the fragments in forward zones, the heaviest con¬ 
centration being about 5° to 15° aft of the shell nose. 
Such patterns are graphical^ portrayed in Figure 
10 (identical with Figure 8 of AMP Note 19). For 
the hypothetical burst surfaces considered, optimum 
fragmentation patterns give heavier concentrations 
of fragments further back from the nose, in a manner 
dependent on the total number of fragments and 
(especially for disks) on the diameter of the burst 
surface. 

For a time-fuzed shell, effectiveness is revealed to 
be more sensitive to increases in the total weight 
of the shell (more fragments in each weight class) 
than for proximity-fuzed shells. On the other hand, 
a more considerable gain is indicated for proximity 
fuzes, rather than time fuzes, from increases in the 
fineness of fragmentation, when the total weight of 
the shell is held fixed. 

n.4.4 AMP Report 185.1R 

This report 11 is primarily concerned with the 
larger scale problems (problems (B) and (C) of the 
introduction) of flak risk and therefore requires solu¬ 
tions for problems (A) and (D) for a large number of 
values of the slant range and elevation. This is the 
first report so far mentioned in which large biases in 
the dispersion of shots had to be considered. In find¬ 
ing the risk to a large group of aircraft from a single 


shot it is obviously impossible to assume that every 
aircraft is at the point of aim; it is necessary to be 
able to estimate the risk t0 each aircraft of the 
formation when the burst distribution is centered at 
some given point (say the lead aircraft of the 
formation). 

Since this report is concerned with problems (A), 
(B), and (C) as well as problem (D), some other 
simplifications had to be made to make it possible to 
take account of the effects of all the new variables 
such as slant range, altitude, and the location of the 
target relative to the point of aim. All computations 
are for time fuzes. The target (as in reference 14) is 
a sphere. A fictitious shell fragment pattern is used 
in which all the fragments are assumed to fly off in 
one main fragment zone of width 20°; it is assumed 
that in that zone the fragment density is constant. 
Hence for any values of the dispersion parameters 
studied in the preceding reports, the solution of 
problem (D) given these for actual shell patterns 
will be more reliable than the solution given in AMP 
185.1R. On the other hand, none of the other reports 
give any information about problem (A). The method 
of computation for this report assumes that the 
probable density of bursts is spheroidal Gaussian 
(equation (3)), but the magnitudes of the standard 
deviations <ri and cr 3 across and along the trajectory 
are taken much larger than in any of the preceding 
studies and are assumed to be determined by the 
(future) slant range from the gun to the target. Some 
account of remaining velocity of the shell is also 
taken as is the influence of altitude on air resistance, 
but the computations do not assume any dependence 
of F(x,y,z ) on the height or speed of the target nor 
any dependence of p(x,y,z ) on the motion of the tar¬ 
get. 

Since this chapter is being prepared by the author 
of AMP 185.1R, it need only be mentioned that the 
whole attitude of this chapter has been influenced 
by the work done and information gathered during 
the preparation of that report. It will be referred to 
again in connection with the discussion (Section 
11.5) of applications of problems (A) and (D) to the 
general problems (B) and (C) which are of more ob¬ 
vious immediate concern to an air force. It may only 
be noted that all the computations carried out for this 
report on problem (A) indicate that for time fuzes 
and for large values of ai and o- 3 , P{a,b,c ) can be 
approximated by an expression very similar to that 
defining F(x,y,z) except that the center is shifted 
forward and the lateral dispersion increased. The 


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186 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


forward shift is due, of course, to the forward spray 
of the fragments from the burst; the increase in 
lateral dispersion is caused by the wide side spray. 
That is to say, if F(x,y,z ) follows the spheroidal 
Gaussian law 

F(x,y,z) = 

then P(a,b,c) can be approximated by a formula of 
the form 

P(a,b,c) = e~ K (a * +bi) /(<n *+^ + < c - "o)*/^ 

where c 0 and e are determined by the shell and its 
remaining velocity as well as being influenced by 
<ti and <73. 

Table 1 shows the variations of accuracy of fire 
with slant range assumed in AMP 185.1R using 



Figure 11. Nomograph showing the relationship 
under service conditions between P 0 , the risk to a 
single aircraft at the point of aim; D, the slant range 
from gun to aircraft; and H, the altitude of the aircraft. 
Nomograph valid for D ^ H. D and H are measured 
in thousands of yards. To use the nomograph lay a 
straight-edge from H on the left-hand scale to D on 
the right-hand scale; read P 0 on the central scale. 


these assumptions. Figure 11 shows a nomograph 
(reproduced from Figure 3s of that report) giving 
the relationship between P Q the risk to a single 
aircraft at the point of aim, H the altitude of the 


point of aim, and D the slant range to the point of 
aim. 


Table 1 . Standard deviations <r 3 along the trajectory 
and (X\ = o- 2 across the trajectory as functions of slant 
range D under “Service Conditions” assumed in AMP 
Report 185.1R. 


D (1,000 yd) 

<j\ (feet) 

<r 3 (feet) 


4 

68 

1,200 


5 

104 

1,130 


6 

158 

1,084 


7 

225 

1.070 


8 

330 

1,120 


9 

450 

1,190 


10 

580 

1,300 


11 

750 

1,420 


12 

940 

1,600 


13 

1,130 

1,860 



11.4.5 Airborne Rocket Fire 

In this study of the risk to a bomber from an air¬ 
borne rocket fired from astern it was expected that 
fire control errors would be small enough that most 
of the risk would come from close bursts, bursts so 
close that the size and shape of the target would be 
important. Hence the conditional probability was 
computed for an actual aircraft (German Ju-88) 
rather than for a spherical target. 

The rocket is assumed to have a proximity fuze; 
the whole study is largely motivated by questions of 
fuze design. This study is more fortunate than OSRD 
738 and AMP Note 19, in that some firing test data 
were available for the proximity fuze T5 of the 4 
in. airborne rocket. 

The technique of calculation of the conditional 
probability function is given in the first 3 of a series 
of reports on the study. The bomber is divided into 
sections and for each section the vulnerable area is 
estimated. For a given point of burst, computations 
from the fragment pattern of the rocket give the 
expected number of hits on each of the vulnerable 
parts. Taking account of the shielding of one part 
by another (from the given point of burst) the total 
expected number m of deadly hits can be found. The 
conditional probability is taken to be 1 — e~ m in the 
usual way. This report also calculates the errors in 
the conditional probability that would arise if the 
actual target with its scattered vulnerable parts were 
replaced by a sphere of the same presented area at 
the center of the actual target. 

Table 2, reproduced from Table 1 of this report, 3 
shows this conditional probability. 


CONFIDENTIAL 












HISTORICAL SUMMARY 


187 


Table 2. Conditional probability that a 4^-in. proximity-fuzed rocket fired from astern will destroy 
a twin-engine bomber. 

Ju-88 equipped with Jumo 211 (liquid cooled) engines and assumed unable to return to base on one engine. 


Distance, (a;) in feet, from tail of 
airplane measured toward nose 

..... 

Probability that burst at indicated point will prevent 
return to base if impact parameter, ( p ) in feet, is 

20 

40 

60 

100 

140 

Aspect angle (0) 

= 0° (bursts directly above airplane) 


-12 

0.136 

0.027 

0.106 

0.075 

0.026 

-2 

0.227 

0.104 

0.244 

0.065 

0.029 

8 

0.253 

0.444 

0.232 

0.073 

0.032 

18 

0.754 

0.458 

0.221 

0.118 

0.080 

28 

0.995 

0.835 

0.591 

0.294 

0.161 

38 

0.989 

0.820 

0.586 

0.272 

0.134 

48 

0.490 

0.349 

0.262 

0.137 

0.070 

58 

0.017 

0.064 

0.077 

0.063 

0.042 

Aspect angle (0) = 

90° ( bursts directly to one side of airplane) 


-12 

0.161 

0.061 

0.033 

0.038 

0.013 

-2 

0.295 

0.065 

0.094 

0.041 

0.016 

8 

0.394 

0.193 

0.156 

0.043 

0.018 

18 

0.382 

0.312 

0.147 

0.047 

0.026 

28 

0.966 

0.563 

0.284 

0.113 

0.058 

38 

0.997 

0.656 

0.386 

0.147 

0.068 

48 

0.637 

0.319 

0.185 

0.076 

0.035 

58 

0.200 

0.041 

0.051 

0.035 

0.019 


The conditional probability is used in a second 
report 4 to find the location of that surface which 
would give the greatest probability of damage if it 
were possible to design a fuze with that burst surface. 

From this material and from records of actual fuze 
performance, in a third report 5 the risk to the bomber 
from a rocket fired from astern was computed in 
terms of the accuracy of aim of the rocket. 

11.4.6 Comparison of Different 
Projectiles 

AMP Study 27 compares the effectiveness of 5-in. 
high-explosive shell, 5-in. shrapnel (both with time 
fuzes), 40-mm and 20-mm HE (both with contact 
fuzes). The computations with time fuzes are made 
in more or less the way described in Section 11.3, 
but the other types of fuzes use modified procedures 
appropriate to their shell types. 6 - 7 

A shrapnel shell acts somewhat like a flying shot¬ 
gun; when the fuze detonates the shell, a number of 
spherical balls are pushed out of the shell case and 
travel forward in the path the shell was following. 
Most of their speed comes from the remaining ve¬ 
locity of the shell but there is some scattering. 
Since the balls are a good ballistic shape and com¬ 
paratively heavy, they retain their effectiveness at 
much greater distances from the burst than do most 


of the jagged fragments from the burst of a high- 
explosive shell. The conditional probability function 
for a shrapnel shell is zero except in a narrow cone 
(of aperture 5° or so) about the nose of the shell. 
This makes the computation for a shrapnel shell 
simpler than for a high-explosive shell where most 
of the fragments come out in a side spray somewhat 
forward of the shell equator. 

The computations for contact-fuzed projectiles are 
simplified from those of proximity fuzes as we dis¬ 
cussed near the end of Section 11.3.3. Those areas 
of the target vulnerable to the particular ammunition 
being used are projected on the x,y plane, giving an 
area U. Then for a hit in each vulnerable region, the 
probability of damage from a hit in that region is 
estimated and the final risk computed from a double 
integral 



u 


where F(x,y) describes the probable density of tra¬ 
jectories in the x,y plane and p(x,y) is the risk from 
a shell on the trajectory passing through x,y. The 
actual computation is done by dividing the target 
area U into sections on which p and F are practically 
constant, computing the risk for each section sepa¬ 
rately, and summing. 

Comparison of the two 5-in. projectiles is com- 


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188 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


pleted with the calculation of the probability of 
damage with a single shot of each kind. To compare 
effectiveness of the other types of projectiles in 
stopping an attack, other factors, such as the rate 
of fire and the range at which effective fire can be 
begun, must also be considered. 



Figure 12. Expected number of hits on each square 
foot of target area presented to the gun as a function 
of the range at which the engagement closes. 7.7-mm 
gun firing 400 shots per minute; opening range is 7,500 
ft, target velocity 250 fps; a is the standard deviation 
of the bullet pattern. 

The earliest AMP study of risk deals with a di¬ 
rectly approaching aircraft and automatic weapons. 
It is assumed that the shots have a normal distribu¬ 
tion about their mean point of impact and that the 
mean point of impact has a normal distribution about 
the target center. The report 1 gives the probable 
number of hits and quick means of estimating upper 
and lower bounds for the number of hits in terms of 
the presented shape of the target and the standard 
deviations of these Gaussian distributions. 


Another study 2 is concerned with the increase in 
risk to a high-altitude bomber due to increase in the 
length of its bombing run. For various lengths of 
bombing runs the risk from both fighter aircraft and 
A A guns is considered. This study was motivated by 
the suggestion that aircraft dropping guided missiles 
might need to continue in steady straight-line flight 
until the missile struck, instead of beginning evasive 
action immediately on releasing bombs. 

Another study compares the expected number of 
hits per square foot of target for four different auto¬ 
matic weapons: 7.7 mm, 13.2 mm, 20 mm, and 40 
mm. This expected hit density is given on a number 
of charts in the report 8 in terms of the rate of fire, 
the range at which fire is begun and stopped, the 
target speed, and the accuracy of fire (assuming a 
constant angular dispersion at all ranges). Such a 
chart is reproduced here as Figure 12. 

A later report 9 compares the effectiveness of the 
20-mm gun, one existing machine gun (caliber 0.50), 
and one projected new machine gun (caliber 0.60) 
for installation in antiaircraft turrets. The com¬ 
parison is made in terms of the accuracy of the guns, 
the rates of fire, and the relative effectiveness of the 
individual projectiles. Comparison of the caliber 0.50 
and the 20 mm on various bases is reasonably reli¬ 
able, but the comparison is quite sensitive to the 
accuracy of fire involved, so conclusions involving 
the caliber 0.60 (with its unknown accuracy) are 
somewhat more tentative. 

A recent study 10 estimates the vulnerability of 
different parts of a very heavy bomber (B-29) from 
different directions and different ammunition types 
(armor-piercing, incendiary, high-explosive, and 
mixed). The numbers given in the tables cannot 
be used directly for other types of projectiles 
or for fragments from a heavy AA shell, but are 
indicative of the relative vulnerability of the 
different parts of the B-29. Table 3, reproduced 
from this report, 10 shows some of these vulnera¬ 
bility estimates. 

The latest study 12 bearing on conditional proba¬ 
bility compares three different types of bombs. For 
each bomb the surfaces are given on which the 
probability is 0.9 or 0.5 of destroying a bomber 
(Ju-88) by a burst of a given bomb on that surface. 
Characteristics of the bombs were taken from 
various TDBS reports (issued by the Chief of Ord¬ 
nance); vulnerability characteristics of the target 
were those used in AMP Study 21 (described in 
Section 11.4.5). 


CONFIDENTIAL 



























APPLICATIONS OF FRAGMENTATION 


189 


Table 3. Vulnerability of B-29 to fighter attack, by components attack from directly ahead. 


Component 

Type of damage 

Equivalent lethal area 
(sq ft) 

Cooperative damage 
assumed required to 
prevent plane from 
returning to base 

7.7 mm 

12.7 

mm 

20 mm 

AP 

I 

AP 

I 

API 

HEI 


Unconditionally lethal 








Gas tanks 

Fire 

0 

6.0 

6.0 

21.4 

25.8 

51.4 


Fuselage 

Serious or lethal wound to both 









pilot and co-pilot 

0 

0 

0 

0 

0 

6.6 



Conditionally lethal 








Each inboard engine 

Engine stoppage 

6.1 

5.5 

12.7 

11.4 

16.8 

15.1 


Associated gas tanks 

Leakage sufficient to stop en- 







Two out of four 


engine; no fire 

0 

0 

3.9 

0 

5.3 

4.5 

engines must be 

Engine plus gas tanks 

Engine stoppage; no fire 

6.1 

5.5 

16.6 

11.4 

22.1 

19.6 

stopped; lethal 

Each outboard engine 

Engine stoppage 

6.1 

5.5 

12.7 

11.4 

16.8 

15.1 

hit on engine or 

Associated gas tanks 

Leakage sufficient to stop en¬ 







associated fuel 


gine; no fire 

0 

0 

3.5 

0 

4.6 

4.0 

tanks sufficient 

Engine plus gas tanks 

Engine stoppage; no fire 

6.1 

5.5 

16.2 

11.4 

21.4 

19.1 

to stop engine 

Fuselage 









Pilot (same for co-pilot) 

Serious or lethal wound; other 







Both pilot and co¬ 


pilot not incapacitated 

1.5 

1.5 

2.5 

2.0 

2.5 

19.0 

pilot must be in¬ 









capacitated 

Each of two sets of ele¬ 









vator controls 

Severing of cables 

0.3 

0.3 

0.8 

0.8 

1.2 

2.2 

Elevator controls 









completely dupli¬ 









cated; both sets 

Rest of frontal area 








must be severed 




Not vulnerable 




Total plane from head on 

Unconditionally lethal 

0 

6.0 

6.0 

21.4 

25.8 

58.0 



Conditionally lethal 

28.0 

25.6 

72.2 

51.2 

94.4 

119.8 



Grand Total 

28.0 

31.6 

78.2 

72.6 

120.2 

177.8 



11.5 APPLICATIONS OF FRAGMENTATION 
AND DAMAGE CALCULATIONS TO 
FLAK ANALYSIS 

Fragmentation and damage studies involve a great 
deal of tedious computation and, hence, are usually 
undertaken only for some serious reason. The prob¬ 
lems that should be studied are raised by two classes 
of people, the shooters and the dodgers. The broad 
questions are these: 

1. What can be done to gun, shell, or director to 
improve the effectiveness of AA guns? Another 
formulation of this question would be: What change 
in effectiveness will result if some proposed change 
in AA equipment and tactics is made? The decision 
to produce proximity fuzes required some sort of 
answer to such a question and led to the studies 
issued as OSRD Report No. 738, 16 and AMP Note 
No. 19. 13 

2. What can be done by equipment and tactics to 
reduce the effectiveness of a given A A defense which 
cannot be avoided? 


The manual Flak Analysis 18 and the report AMP 
185.1R 11 were motivated by this latter question. 
Naturally, the whole problem of effectiveness of 
bombing operations is more than a problem of re¬ 
ducing losses. It is an intricate problem of balancing 
bombing results against losses from flak, fighters, and 
operational difficulties. 

The principal methods of reducing the total risk 
from AA fire may be roughly classified as follows: 

1. Minimizing the number of shells fired per bomber 
by 

(a) Reducing the time of exposure to AA fire 
through increased ground speed, reduced bomb¬ 
ing runs, and carefully planned approach and 
withdrawal. 

(b) Attacking AA batteries in advance of bomb¬ 
ing missions. 

(c) Devising formations and planning the spacing 
between formations so that fewer A A guns, sta¬ 
tionary and mobile, per attacking plane can be 
effectively brought into action. 

2. Decreasing the number of aircraft required for 


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THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


bombing attacks by an increase in bombing efficiency. 
Note that this might involve tactical recommenda¬ 
tions at variance with other methods of reducing 
flak risk. A judicious balancing of conflicting prin¬ 
ciples is most important. 

3. Reducing risk of damage per aircraft per shell by 

(a) Measures intended to decrease the accuracy 
of AA fire (evasive flying, night attacks, high alti¬ 
tudes, radio countermeasures, attacks on AA 
batteries). 

(b) Improvements in bomber design (distribution 
of vulnerable parts, protective shielding, etc.). 

Fragmentation and damage calculations do not 
appear directly in many of these considerations, but 
all of the measures intended to reduce the accuracy 
of enemy fire influence the values of any parameters 
used in the calculations. Hence in any such compu¬ 
tations it is necessary to know the circumstances for 
which the computations are to be made. Let us dis¬ 
cuss some cases when these computations are useful 
in furnishing basic information. 

11.5.1 Flak Charts 

To find the safest direction of approach and with¬ 
drawal from a gun-defended target requires some 
measure of the total risk accumulated by a group of 
aircraft during its run over the target area. This re¬ 
quires a means of estimating the risk on each shot 
and the probable rate of fire, and accumulating the 
risk along the given bomb-run. 

For a single aircraft at the point of aim, the prob¬ 
ability P Q of damage to that aircraft can be computed 
by means of the formulas of Section 11.3 in any 
tactical situation where the accuracy of fire is known. 
If the dependence of accuracy on slant range, alti¬ 
tude, and other variables can be determined, then it is 
possible to compute P 0 for a given target aircraft in 
terms of its position relative to the gun. From the 
path of the target and the rate of fire of the gun, it is 
then possible to find the probable location of suc¬ 
cessive points of aim along that path, and compute 
the risk for each shot from its position. Then the sum 
of these risks gives the total risk for the course. 

If it is assumed that the risk is negligible except on 
the bomb run, the computation can be simplified 
somewhat and the results shown on charts (called 
flak charts) which can be used to compute fairly 
rapidly the risks from a given gun defense. One 
chart is to be made for each of a selection of alti¬ 
tudes, say every 5,000 ft. 


1. For a given altitude and gun draw a circle 
whose radius is the horizontal range within which 
the given gun can fire shells to the given altitude. 

2. A number of parallel crossing courses, say at 
intervals of 500 or 1,000 yd, are drawn across this 
circle; each of these courses is divided into intervals 
of some convenient length, say L yards. 

3. From the speed of the aircraft, the time be¬ 
tween shots, and the time-of-flight curves for the 
given gun, the expected number of bursts in each 
interval can be computed. The formula is 

L AT 

expected number of bursts = — 4 - — > 

D n 

where L is the length of the interval; 

n is the number of seconds between shots; 

D is the distance the aircraft travels in n sec¬ 
onds (D and L must both be measured in 
the same units); 

AT is the time of flight (in seconds) of the shell 
from the gun to the beginning point of the 
interval minus the time of flight to the end 
point of the interval. 

Since the distance D in the formula is related to the 
speed v of the aircraft by the equation D = nv , we 
see that when L/nv is large compared with AT/n, 
i.e., when the aircraft is not traveling too rapidly so 
L/v is large compared with AT, the first term con¬ 
tributes most to the expected number of shots in the 
interval and that number of shots is, therefore, ap¬ 
proximately inversely proportional to the speed. A 
flak chart is drawn for some fixed speed and the 
values of the risks computed for it are then corrected 
for true speed by multiplying by the ratio 

speed used in constructing flak charts 
true ground speed 

This correction is not precise for large changes of 
speed due, for example, to 100-mph winds at the 
target. 

4. The average probability of damage to the air¬ 
craft from a burst in each interval is estimated by 
calculating P 0 at the center of the interval. 

5. The total risk in the entire interval is the 
product of the expected number of bursts times the 
average risk per burst. 

6. These risks are accumulated along each cross¬ 
ing course to show how the total risk from the gun 
increases as the aircraft flies along that part of the 
crossing course within range of the gun. 

7. Contour curves for this risk are drawn to com- 


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191 



Figure 13. Flak chart for a single aircraft at altitude 8,000 yards with ground speed 293 miles per hour under service 
conditions (described in Table 1) with continuously pointed fire at the rate of 12 rounds per minute. The unit of risk is 
the risk to a single aircraft at 8,000 yards altitude and 10,000 yards slant range. 


plete the flak chart. The number on each contour 
curve represents the risk (in some convenient units) 
due to flying along a crossing course in the given 
direction of flight up to that contour curve. Figure 
13 shows a flak chart prepared in this way from the 
values of P 0 given in AMP report 185.1R. 11 The gun 
is assumed to be the U.S. 90-mm AA gun firing time- 
fuzed shells. The heading “service conditions” de¬ 
notes the standard of accuracy used. The errors 
along and across the trajectory are given as functions 
of slant range by Table 1 of Section 11.4.4. The unit 


of risk is the value computed for P 0 under service 
conditions at 10,000 yd slant range and 8,000 yd 
altitude. For more details see reference 11. 

To use a flak chart to select the safest course for a 
bombing run, the chart is used with a map drawn to 
the same scale showing the target to be bombed and 
the batteries defending it. The flak chart itself is 
made on a transparent film so that the map can be 
read through it. For any proposed bombing run, 
drawn on the map, the center of the flak chart is set 
on each battery in turn with the direction of flight 


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192 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


arrows pointing along the bombing run. The risk at 
the beginning of the run is subtracted from that at 
the end (these are read from the contours of the flak 
chart) and this difference multiplied by the number 
of guns in the battery to give the risk to the aircraft 
on that run from that battery. The process is re¬ 
peated for all the batteries defending the target and 
the values from all batteries added together. The 
result is multiplied by the correction factor for ground 
speed to get the risk for that bombing run. (A re¬ 
finement of this technique estimates the position of 
first accurate burst and uses that point instead of the 
beginning of the bomb run in computing the risk.) 
Fully detailed explanation can be found in a refer¬ 
ence report. 18 



Figure 14. Relative hit expectancy as a function 
of the direction of attack. 


The expected wind over the target has two effects 
on the computation; it moves the bomb release point 
and changes the speed so that the bombing run on a 
given target with wind is in a different position and 
of different length (on the ground) than it would be 
without wind. Hence the computation with wind 
uses a different bombing run and corrects by a dif¬ 
ferent speed factor at the end of the process than the 
computation without wind. The general effect is that 
the increase of speed from going downwind into a 
well-balanced AA defense makes it much safer than 
going upwind into the same defense; if the wind 
speed is high, the wind direction is often the most 
important factor in determining the safest route into 
and out of the target area. 


This computation is performed for each of a num¬ 
ber of directions from the target, say every 30°, and 
the risks for bombing runs from these directions are 
displayed graphically in some manner to show safest 
route in and out. Two sample methods of representa¬ 
tion are shown in Figures 14 and 15. The first is a 
polar coordinate diagram showing the relative risk 
for attack in each direction as a distance in that 
direction. The second, a “flak clock,” shows only 
preferred directions of entry and withdrawal with 
the order of this preference. 



Figure 15. “Flak clock” showing preferred direc¬ 
tions of attack and withdrawal with order of preference. 


n.5.2 Flakometers 

From flak charts it is possible to estimate the de¬ 
pendence of risk during a bombing run on the di¬ 
rection and altitude of the run. To isolate the effect 
of altitude we construct a figure, called a flakometer, 
showing the typical accumulation of risk at different 
altitudes. 

We start with a given bombing unit. In preparing 
flak charts for that bombing unit we include the 
effect of altitude on the true airspeed. We use again 
the X,Y,H coordinate system of the preceding sec¬ 
tions to locate the position of the target aircraft rela¬ 
tive to the gun. To estimate the risk at a given 
altitude we take the following steps: 

1. On the flak chart for the given altitude draw 
crossing courses parallel to the Y axis spaced 1,000 
yd apart. 

2. If the bombing unit flies along one of these 


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APPLICATIONS OF FRAGMENTATION 


193 



Figure 16. Flakometer. Single aircraft, service conditions. This figure shows the risk to a single aircraft from a line 
of guns spaced evenly one gun per 1,000 yards along a line at right angles to the line of flight. The flakometer also shows 
the relative risk in passing over a gun or battery whose distance from the course of the aircraft is not known. 


crossing courses to the point ( X,Y,H ), the risk ac¬ 
cumulated since the unit came within range of the 
gun can be read from the flak chart; call this risk 
F(X,Y,H). 

3. The value of Y determines which crossing course 
is involved; the crossing courses selected in step (1) 
are determined by the values Y m = 1,000m where 
m = 0, ±1, ±2, • • •. If we wish to suppress the 
effect of the particular crossing course involved, for 
each X and H we add all the values F(X,Yi,H ) to get 
a quantity which we will call 2(X,H). 

4. A flakometer is a set of contour curves for the 
function 2(X,H). 

This sum 2(X,#) of the risks on a collection of 
crossing courses can be regarded rather simply as 
the risk to the bombing unit from a line of guns spaced 
1,000 yd indefinitely far in either direction along the Y 
axis. As the altitude H decreases and X is held 
fixed, 2(X,#) increases from a combination of three 
effects. First, as H decreases more crossing courses 
come within range of the gun so there are more 
terms in the sum making up 2(X,H). Second, for 
each X slant range decreases and hence F(X,Y,H) 
increases because larger values of P go into it. 


Third, speed decreases with H so the unit has been 
in range longer at low altitudes, so again for each X 
and Y, F(X,Y,H) tends to increase when H de¬ 
creases. 

Figure 16 shows a flakometer constructed under 
the same assumption about accuracy of fire as was 
the flak chart of Figure 13. In constructing the 
flakometer, the variation of ground speed with alti¬ 
tude was also prescribed. 11 

n.5.3 Group Probability Factor 

When a number of aircraft are flying together, a 
shell aimed at one aircraft may damage another; 
hence the risk to a group of aircraft in a given posi¬ 
tion from the gun is greater than the risk to a single 
aircraft. However, the risks to the various aircraft 
in the group are not all the same; presumably the 
farther an aircraft is from the point of aim, the safer 
it is, so generally, the risk to a group of n aircraft is 
not just n times the risk to any particular one of 
them. To compute the risk to the whole group it is 
necessary to estimate the risk to each aircraft in the 
group and sum these risks. 


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194 


THE RISK TO AIRCRAFT FROM HIGH-EXPLOSIVE PROJECTILES 


For time-fuzed shells this is a relatively simple 
matter. Given the parameters ai and a 3 the compu¬ 
tation of risk to a given aircraft of the group can be 
carried out when its position relative to the point of 
aim is given. This is done from the formula for 
P(a,b,c), equation (16), or by some approximate 
method. Then summing the risks to the individual 
aircraft gives, approximately, the risk to the whole 
formation; the only error is due to shielding of one 
aircraft by another and this can probably be neg¬ 
lected. 

For proximity fuzes the risk to a group of aircraft 
is not a simple combination of the risks to the indi¬ 
vidual aircraft. The reason for this is clear. For a 
given shell there is a surface in front of each aircraft 
on which the shell would burst if that target were 
alone and the shell happened to pass through that 
surface. When the aircraft are grouped together the 
risk to a single aircraft in the group is no longer the 
risk from a burst on the surface surrounding the 
target but is the risk from a burst on any burst sur¬ 
face surrounding any aircraft. That risk, at least for 
tightly packed formations, may be much smaller or 
much larger than the risk when that aircraft is iso¬ 
lated and has only its own burst surface to operate 
the fuze. Shielding will have a very strong effect 
with proximity fuzes, which it does not have with 
time fuzes. This shielding is not really a shielding of 
fragments but of bursts; it will be due to the fact 
that the bursts will occur before the shell reaches the 
first aircraft near its path and hence the bursts will 
be farther from those aircraft away from the gun. 
(With time fuzes, the bursts are as likely as not to 
be among the aircraft farther from the gun.) If, on 
the other hand, no aircraft happen to be shielded, 
the risk to each aircraft in the group is greater than 
if it were alone. 

If we divide the risk to the whole group by the 
risk P 0 to a single aircraft in the same tactical situ¬ 
ation, we get a number, called the group probability 
factor of the group, which depends on the accuracy 
of fire, the type of fuze, the number and arrangement 
of aircraft in the group, and (for proximity fuzes, es¬ 
pecially) on the angle between the direction of 
flight of the target and the trajectory of the shell. 


If a flak chart is to be built for a group of aircraft 
instead of a single aircraft, then the steps of Section 
11.5.1 can be carried out with only one change. The 
probability P 0 in step (4) is to be multiplied by the 
group probability factor of the group, computed at 
that same point in space, to get the risk to the group. 


11.6 CONCLUDING REMARKS 

The preceding sections have discussed methods de¬ 
veloped during World War II for comparing the 
effectiveness of different tactics and equipment in 
increasing or reducing risk to aircraft from antiair¬ 
craft fire. These methods, of necessity, involve many 
idealizations of more or less doubtful validity. The 
degree to which the results conform to actual, prac¬ 
tical situations is subject to improvement in two 
basic, interrelated ways: (1) by modifying the hy¬ 
potheses in the direction of greater realism and 
(2) by accumulating a larger body of reliable ex¬ 
perimental data bearing on all phases of the problem. 
Insofar as available experimental data are inade¬ 
quate, there is comparatively little immediate prac¬ 
tical value in refinements of the theory. 

One important question for combined theoretical 
and experimental investigation is the distribution in 
space of shell bursts from an A A battery, as de¬ 
termined by the random and systematic errors of the 
director-computer-gun systems. The assumption of a 
Gaussian distribution about the target as mean, with 
independent errors in range, elevation, and azimuth 
is of questionable validity. 

The practical desirability of any investigation of 
the sort just suggested, involving specific pieces of 
equipment, can be assessed in terms of (1) probable 
abandonment or modification of such equipment and 
(2) possible development, in the course of the analy¬ 
sis, of methods applicable to other problems. From 
the latter viewpoint, it was considered desirable to 
stress methods rather than details in the present 
chapter; all the more so since unusually rapid changes 
in equipment and tactics can be expected to accom¬ 
pany further development of atomic power. 


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PART IV 


GENERAL 


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Chapter 12 


COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


12.1 INTRODUCTORY REMARKS 

rom the first, it must be clearly emphasized 
that it is not the purpose of this chapter to pre¬ 
sent a general theory of air warfare; but rather to give 
some tentative, and very incomplete and preliminary, 
ideas of what such a theory could and should com¬ 
prise; to give some of the arguments for and against 
attempting to construct and use such a theory; and 
to indicate how certain activities of the Applied 
Mathematics Panel and of other agencies relate to a 
scheme for a broader and more inclusive analytical 
approach to the problems of air warfare (and of war¬ 
fare in general). 

The ideas herewith presented as to the scope, 
nature, and procedures of a general theory of air 
warfare must of necessity be wholly tentative. For 
such an overall analysis has not, so far as known, 
ever been attempted. We are dealing here with a 
tremendously complex situation, which not only has 
a bewildering number of technical elements, but is 
also basically affected by a host of practical non¬ 
technical considerations of a very general character, 
such as Army-Navy-Air Force organization and 
policy, attitude of high command toward civilian 
technical assistance, attitude of field command 
toward innovations in procedure, conditions which 
attract and hold scientific personnel of high caliber, 
public support of effective measures for military 
preparedness. The precise character of such studies 
as are discussed herewith would also be profoundly 
affected by new weapons and the new tactics and 
strategy which they permit and demand. These 
factors — political, organizational, and technical — 
are obviously impossible to analyze or anticipate at 
this time, and accordingly the ideas presented here¬ 
with must necessarily be vague and general when they 
attempt any accuracy or permanency, and tentative 
and illustrative only, when they attempt to be 
specific. 

The author of these comments, moreover, makes 
no pretense to any professional knowledge of military 


science. Thus he runs the risk of using phraseology 
which may seem naive or even ridiculous to the pro¬ 
fessional soldier. But we cannot get forward with 
this statement unless that risk is taken. All of what 
follows is thus to be considered as protected by a 
blanket apology for blunders of terminology, of 
understanding, or of fact. 

12.2 OFFENSIVE AND DEFENSIVE AIR 
WARFARE 

Offensive air warfare consists primarily of the un¬ 
forced application of bombs (HE, incendiary, atomic, 
etc.), guided missiles (jet or rocket propelled or glid¬ 
ing; target seeking or externally controlled), ordinary 
projectiles (ranging in caliber from 0.50 in. to 75 mm), 
or rockets for the purpose of interfering with or 
destroying: 

1. Uniformed enemy combatants. 

2. Enemy military materiel (planes, tanks, guns, 
radar installations, ships, subs, fuel and ammunition 
stores, etc.). 

3. Enemy communication, and transport in enemy 
countries. 

4. Production facilities of enemy. 

5. Enemy morale. 

It also has the further purpose of destroying any and 
all enemy defenses (fighter planes, antiaircraft guns, 
radar, controlled missile sites, etc.) against such 
offensive attacks. 

Defensive air warfare consists of forced applica¬ 
tions of the weapons of the air arm against enemy 
agencies which seek to carry out direct attacks on 
us. Among such agencies are: enemy aircraft, con¬ 
trolled missiles and their launching sites, ships ap¬ 
proaching our shores, etc. 

It will be noted that offensive warfare has been 
here defined as the use of bombs, rockets, etc., there 
being no explicit mention of own aircraft. Except for 
the possible suicidal (or remotely controlled) use of 
aircraft as a defensive ramming device, the aircraft 
itself is primarily a means of transportation which, 



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197 


198 


COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


with great speed, over long distances, and often un¬ 
seen, brings to the effective place the bombsight and 
bomb rack, the rocket launcher, the gun, etc., which 
then attacks the enemy. 

There has been no mention of the distinction be¬ 
tween strategic and tactical offensive air action. The 
above list of numbered items starts with tactical 
actions, and ends with strategic actions. The dis¬ 
tinctions between tactical and strategic action refer 
primarily to the time factor and to specific operations 
as contrasted with general preparation and direction 
— the actions which at once affect the military 
strength of the enemy of some particular place being 
tactical; those which affect the military strength of 
the enemy in general and after some time delay being 
strategic. 

12.3 INTERRELATIONS 

The brief remarks made in the preceding section 
concerning offensive and defensive air warfare were 
made for the single purpose of emphasizing that 
there is, in general, no such thing as isolated warfare. 
A modern war involves the productive capacity of a 
whole people, its ability to transport materials of war, 
its total capacity to strike by land, sea, and air. The 
part of this whole struggle which comes under the 
label “air warfare” is enormously larger and in¬ 
calculably more important than in the past. But the 
part called air warfare is in most cases (and in all 
large and important cases) inextricably interrelated 
with all the other parts of the total effort. It is as 
unreasonable and self-defeating to attempt an ex¬ 
clusive theory of air warfare as it would be for a 
physician to look through a peephole at a patient’s 
forehead and try to diagnose the cause of his head¬ 
ache. 

Thus, the first main point is: 

There cannot he any such thing as a theory of air 
warfare , unless one is in reality prepared to study war¬ 
fare in general. 

That naval forces, infantry, artillery, air forces, 
coastal defense, production, food, manpower, critical 
materials, training, transportation, time, etc., are all 
interrelated] y involved in any broad strategic plan 
is fairly obvious. Consider, for example, a strategic 
plan to reduce the level of machine tool production 
in an enemy country. What proportion of our own 
productive strength, money, manpower, etc., can we 
afford to expend in order to effect various levels of 
reduction of enemy machine tool production within 


certain stated times? The ramifications of this ques¬ 
tion, the cyclic intertwining of the manifold military 
and economic factors, are obvious. 

But consider some much more detailed and specific 
questions — ones which might easily face a colonel 
at Wright Field, for example. A new bombsight is 
under development, intended for high-altitude radar 
bombing. Suppose the research and development 
agency reports that: 

1. The mil error can be reduced from 50 mils to 40 
mils by adding 30 lb to the weight; or to 20 mils by 
adding 100 lb; or to 10 mils by adding 500 lb. 

2. The mil error which can be achieved through 
essentially perfect operation can be reduced by a 
factor of two by increasing the complexity of the 
sight and the complexity of operation (hence also 
increasing the cost, time to produce, time to train, 
maintenance difficulties, etc.). 

3. The bombsight can be made to operate without 
the necessity of a straight bombing run by further 
increasing its weight by 300 lb. 

What factors should enter into the analysis on 
which the colonel bases his recommendations? He 
cannot possibly himself know all the necessary 
things, but somehow he should certainly bring to 
bear on these questions a wide and precise knowledge 
of the probabilities of bombing accuracies; the 
logistics of the theatres in which these sights are to 
be used; the nature of the enemy targets; all the vast 
field of terminal ballistics; the importance of the 
time factor (which means war plans, among other 
things); the psychology and physiology of operation 
of bombsights; the selection and training of bom¬ 
bardiers; accessibility of qualified personnel; the basic 
cost of accomplishing the same objectives otherwise; 
the present and potential future effectiveness of the 
enemy’s fighter attack against our bombers; etc. 

Thus an apparently detailed question concerning 
the design of one specific piece of equipment cannot 
be answered logically or conclusively except through 
an analysis which takes into account an almost 
frightening array of factors. 

The “practical” man will, at this point, wish to get 
in an emphatic disclaimer. He will wish to point out 
that these questions have to be answered in a reason¬ 
able time, that it is not possible to furnish each 
Wright Field colonel with a board of military Ein¬ 
steins, that “ judgment” “ experience ,” “common 
sense” perhaps even “ horse sense ” are more im¬ 
portant here than a lot of fine-spun theory. 

The practical man is asked to exert an almost super- 


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NATURE OF A GENERAL THEORY 


199 


human restraint, and read on. For it is hoped that his 
fears and objections will receive something of an 
answer before this chapter is concluded. To match 
his restraint, the author agrees to restrain himself and 
to cite, from his experience of the last five years, 
only allegorical and anonymous examples of the kind 
of decisions which, in such problems, sometimes re¬ 
sult from "common sense” and “experience,” di¬ 
vorced from technical knowledge and analysis. 

12.4 NATURE OF A GENERAL THEORY 

In order to expose even the rudimentary ideas of 
a general theory it is unfortunately necessary to 
face a certain amount of complication. That is un¬ 
avoidable, since the complication is an inherent and 
characteristic aspect of such problems. And to enable 
us subsequently to talk without the constant repeti¬ 
tion of long qualifying clauses, we must introduce just 
a little notation. 

Suppose that a certain operation 0 is under con¬ 
sideration, the plan P for this operation becoming 
specific and definite only after a number n of decisions 
D n have become definitely settled. 

Thus the operation might relate to some specific de¬ 
tail in the design of a piece of equipment — say the 
maximum slewing speed of an aircraft turret. In this 
case the plan P is determined when one single de¬ 
cision Di is made definite, that decision consisting of 
assigning a definite number of mils per second to the 
maximum slewing speed. To take another very 
simple illustration, if the operation 0 involved the 
single question as to whether or not the turret con¬ 
trols utilize so-called aided tracking, then the plan 
P consists of the single decision Di of deciding “Yes” 
or “No.” In any such case as this last one, the de¬ 
cision, although consisting of a choice between yes 
and no, can be given a numerical form (and thus 
made similar to the previous case) by arbitrarily 
calling 1 the equivalent of yes, and 0 the equivalent 
of no. Thus in both the illustrations given so far, the 
plan becomes definite when numerical values are 
assigned to the decision variables D n . 

In a case of slightly greater complexity, the oper¬ 
ation 0 might have a plan P which involves two de¬ 
cisions, namely: D h at what altitude shall we carry 
out bombing attacks on a particular target; and D 2 , 
shall we carry these out by day? It is assumed here 
that many of the other variables of the situation 
(types of planes, types of bombs, etc.) are fixed by 
considerations not under our control. 


Here the plan P becomes definite when a numer¬ 
ical value is established fqr A (say 30,000 ft), and 
when yes or no (that is, 1 6r 0) is determined for A, 
so that day or night bombing is settled upon. 

In a case of iliuch more substantial complexity, 
the operation O might involve the optimum use of a 
given sized fleet on B-29 bombers operating out of 
the Marianas against Japan. But even now the 
problem is not nearly as broad as it might be, for it 
presupposes aircraft of certain definite character¬ 
istics, operating from a given set of bases; whereas a 
broader problem would include a consideration as to 
whether the attack should be made by inhabited air¬ 
craft (and what types of aircraft), remotely con¬ 
trolled aircraft, guided missiles, etc. But even in this 
somewhat restricted problem it is clear that the plan 
P for the operation now includes a large number of 
decision variables D n Avhose values must be de¬ 
termined. 

Thus, to indicate only a few, one would have to 
assign priorities to a set of targets (thus involving 
numerical values for a set of decision variables, one 
of which corresponds to each target under consider¬ 
ation) . One would have to settle on a flight plan, this 
in turn breaking down into a large number of indi¬ 
vidual decisions, each of which must be definitely 
determined. This flight plan, among many other 
items, involves decisions as to the formations which 
will be flown en route, when fighter opposition ap¬ 
pears, when flak is present, for the bombing itself, 
and on the return trip. All of these matters can be 
systematically set down in such a way that they are 
definitely settled by assigning numerical values to 
various decision variables D. Evasive tactics must be 
included in this flight plan, and in considering vari¬ 
ous formations and tactics it is clear that one is 
forced into a detailed consideration of the accuracy 
and effectiveness of the defensive fire of the bombers, 
their vulnerability to enemy attack by fighter planes, 
flak, rockets, air-to-air bombing, self-inflicted dam¬ 
age, etc. Questions of fuel consumption, wear on 
aircraft, crew morale, maintenance, training, psy¬ 
chological and physiological limitations on gunners, 
bombardiers, pilots, etc., must be considered. 
Weather, radar navigation aids, and radar bombing 
aids are intimately involved. Probable damage to 
own aircraft from flak, rockets, fighter attack, etc., 
involve complex and subtle studies of the vulner¬ 
ability of aircraft components and personnel, frag¬ 
mentation characteristics of enemy shells, fire control 
accuracies, etc. Effectiveness of the bombing involves 


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200 


COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


all the difficult probability and statistics of bombing 
accuracies, and the vast array of factors which enter 
into terminal ballistics. The enemy defenses of all 
sorts and against all types of attack must be esti¬ 
mated as best one can on the basis of past experience 
and Intelligence (the past experience, incidentally, 
not being worth much if it consists merely of the 
understandably emotional reaction of crew members: 
provision must be made for first hand analysis in the 
field and at central locations by technically compe¬ 
tent personnel such as the Operations Analysis 
Sections, a and at central correlating and analysis 
bureaus such as did not really exist during World 
War II. 

Thus in this third illustration of an operation 0, 
the plan P clearly involves a very large number of de¬ 
cision variables D n which must be determined in 
order to decide the plan. And the brief description 
given above indicates another feature of such prob¬ 
lems— a feature which we have not mentioned as 
yet, but which is essential. 

It is, in fact, clear that the decision variables D n 
themselves depend upon a large number of further 
and more basic variables, such as the fuel con¬ 
sumption of a B-29 at given load, speed, and alti¬ 
tude; the percentage of fragments larger than, say, 
one ounce in weight, which leave a certain bursting 
AA shell in certain angular directions; the proba¬ 
bility that two .50-caliber slugs in a certain area of a 
fighter plane cause a fire; the probable value in mils 
of the harmonization error of certain guns of a B-29 
when flying in bumpy air; the probable number of 
days per month (in a given season of the year) that a 
certain enemy target will be visible from 14,000 ft 
and not visible from 30,000 ft; etc. These basic vari¬ 
ables, which we will call Vn (thus indicating that 
there are N of them) affect the decision variables 
D n , but are distinct from them. 

Certain of the basic variables V are at our dis¬ 
posal, in that we can control their values. Others are 
under the control of the enemy (such as, for example, 
the probability that enemy fighters attack at a cer¬ 
tain place and time). Others, such as weather, are 
under the control, so to speak, of nature. 

It is clear that the relations between the decision 
variables D and the basic variables V are very 
complicated. These two sets of variables are all tied 
together in all sorts of complicated ways. Some of the 
decision variables D are not logically independent, 


a See, for example, reference 42. 


for assigning a value to one’may involve restricting 
certain of the basic variables to values, or ranges of 
value, which in turn may effectively limit and may 
even specifically determine the value of other de¬ 
cision variables. This is, of course, merely the formal 
mathematical equivalent of the obvious fact of ex¬ 
perience that often one decision commits a person 
to certain further decisions. 

Among the variables which play important roles 
are the quantities, say N j} which designate the num¬ 
bers of certain types of weapons which we or the 
enemy may have available at certain times and 
places. These variables may be decision variables in 
some problems, and basic variables in others. In 
strategic calculations one is interested in the values 
of Nj over fairly long time intervals — one or two or 
even more years. In tactical calculations, one is 
interested in the values of Nj at specific places, and 
over a shorter future interval of, say, a few weeks or 
months. 

We have progressed far enough in a preliminary 
discussion so that we can now describe the essential 
and central feature of a general theory of air warfare. 
The central feature is obvious and simple. It consists 
merely of saying that the plan P should, in fact, be 
characterized by that particular set of decisions D 
which is worth most to us, namely the set which gives 
us the largest margin of profit — the largest excess 
of return over cost. 

The words profit, return, and cost do not, of course, 
refer to worth as measured in dollars. They refer 
rather to what may be called military worth. On the 
profit side of the ledger one takes account of the 
destruction or other sort of harm which is imposed 
on the enemy. If the concept of military worth is 
developed along relatively decent lines, destruction 
and harm to the enemy will be counted as the more 
valuable, the more definitely and promptly this 
destruction and harm contribute toward the success¬ 
ful termination of the conflict. Longer range con¬ 
siderations involving possible future conflicts, how¬ 
ever, cannot safely be entirely neglected, as is well 
illustrated by the decision to use the atomic bomb 
on Hiroshima and Nagasaki. On the cost side of the 
ledger one must take account of our necessary ex¬ 
penditures of labor, time, critical materials, money, 
materiel, personnel, etc. In some simple cases it may 
be possible to treat military worth as closely analo¬ 
gous to man-hours of labor. In other cases nothing so 
simple will suffice. 

Military worth, as the phrase is here used, is 


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201 


closely related to the general concept of utility in 
economic theory. And the reader is warmly urged to 
read the discussion of a numerical theory of utility 
given (on pages 15 to 29 and elsewhere) in “Theory 
of Games and Economic Behavior” by John von 
Neumann and Oskar Morgenstern. 1 This pioneering 
and brilliant book is, it should be pointed out, con¬ 
nected in a most important way with the viewpoint 
here being presented, for it develops a large part of 
the mathematics necessary for theories of competi¬ 
tive processes. 

But assuming that there has been developed a sat¬ 
isfactory concept of military worth (for which we 
will use the notation MW), then the essential pro¬ 
cedure of a general theory of warfare is to determine , 
for any operation 0 a plan P, whose values of the de¬ 
cision variables D n maximize the military worth MW. 

Let us try to make this situation somewhat more 
clear and definite. Suppose for the moment that a 
satisfactory concept of military worth has been de¬ 
veloped, that techniques have been elaborated for 
computing, relative to a given operation 0 numer¬ 
ical b military worths of various plans P, that is to 
say, of various choices of the decision variables D n . 
In fact, let us suppose that there has been constructed 
for this operation 0 a great Tactical-Strategic Com¬ 
puter [TSC]. This computer has one main output 
dial — the one which displays the military worth. 
It has a second set of dials, one for each of the decision 
variables D n . And it has a third set of dials, one for 
each of the basic variables Vn. 

Suppose we seat ourselves, in imagination, before 
this tactical-strategic computer and play with its 
controls. For this process will illustrate, and perhaps 
make more clear, some things said above. 

We start by setting in the appropriate values to 
those basic variables V which are under enemy con¬ 
trol, and those basic variables V which are under 
nature’s control. In some cases (as, for example, the 
setting time for a well-known enemy antiaircraft 
director) these may be inserted as definite fixed 
values. In other cases (as, for example, weather con¬ 
ditions over certain enemy targets) the computer 
must be equipped to accept parameters which char¬ 
acterize the statistical behavior of the quantity in 
question. In still other cases (as, for example, prob- 

b Quantitative significance is not required of the scale of 
values of MW. Only linear order is important. Since both 
cost and profit have been taken into account, one set of 
decision variables is unambiguously better than a second set 
just provided the first set leads to a larger value of MW. 
How much larger is of no importance. 


able mil accuracy of enemy fire from flexible guns in 
aircraft) the computer must be equipped to accept 
a pair of values, setting lower and upper limits to the 
quantity in question; in the computing process which 
will take place presently, this particular basic vari¬ 
able will be allowed to vary over this estimated range. 
Other basic variables of this same type will similarly 
vary, but with constantly changing phase relations, 
so that within a short time the TSC has in fact taken 
into account all combinations of values of those 
basic variables for which it is necessary to estimate 
a range. 

Having set in the “enemy” and “nature” basic 
variables (but leaving many of the other basic vari¬ 
able controls as yet untouched), we now begin to 
twiddle the decision variable dials. We may, for 
example, have in mind a certain set of the decision 
variables — one suggested, for example, by the Air 
Forces Board, or by Eglin, or by some general with 
an astronomical number of stars. We start to set 
these values of D n in, one after another. As we do, the 
inner calculating mechanisms are set in operation, 
and values (or in some cases indicated ranges of 
values) begin to appear on these basic variable dials 
which we did not initially set. For example, if we set 
onto one of the decision variable dials a value which 
corresponds to the choice of a 5-in. rocket of a certain 
type, then the basic variable dials corresponding to 
fragment, angular, and mass distribution character¬ 
istics, move over to proper settings. If we set a de¬ 
cision variable dial to correspond to a certain sighting 
system, then the basic variable dials corresponding 
to probability of damage, etc., indicate ranges which 
correspond to this system. 

It is perfectly possible that, after setting in a cer¬ 
tain number of values of decision variables, the 
machine is found to be locked when one tries to set 
in another decision variable. This means that the 
decisions already made in fact determine the remain¬ 
ing decisions. The complex interrelationships, via all 
the basic variables, is such (and perhaps quite un¬ 
expectedly such) that the various decisions are not 
really independent. In this case, one pushes a special 
button, and all the remaining decision variable dials 
move automatically into their necessary positions, 
thus showing what further decisions we are perforce 
committed to. 

In case, however, the decisions all prove to be in¬ 
dependent, one continues until all their values have 
been set in. At this juncture (or very shortly there¬ 
after, corresponding to some time delay required by 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


the computing mechanisms) the military worth dial 
lights up and displays the numerical value of MW. 
Only now do the really interesting things begin to 
happen. First of all, one watches the value of MW for 
a time during which the computer is trying out all 
combinations of those basic variables for which 
ranges, rather than specific values, were set in. If 
the MW varies widely during this period, one may 
possibly conclude that, before attempting to proceed, 
further research is necessary in order to delimit these 
basic variables to more narrow ranges. One may play 
with specific ones of these basic variable dials to 
determine which one or ones it is whose uncertainty 
contributes principally to the uncertainty in MW. 
Similarly for the basic variable dials on which have 
been set specific numbers (or statistical parameters) 
one can shift them one at a time to see whether the 
MW is sensitively or sluggishly increased thereby. 

But of greater probable interest is the behavior of 
the MW dial when one alters the values set into the 
decision variable dials. For then one is changing the 
military plan for the operation in question , and is ob¬ 
serving directly whether the change is really for the 
better or for the worse. To seek the optimum plan, one 
would set into operation a mechanism which (again 
with changing phase relation) shifts all the decision 
dials through cycles of accessible values, the resulting 
values of MW being recorded so that the maximum 
-can be located and the corresponding set of optimum 
values of the decision variables D n determined. 

It should be admitted at once that so complete 
and so formally mechanized an analytical procedure 
doubtless lies far in the future. The speed and flexi¬ 
bility of modern digital and analogue computing de¬ 
vices (particularly those which operate at electronic 
computing speeds and have very flexible control 
systems), and the accuracy and stability of modern 
multistage servo systems (permitting great freedom 
in the use of feedback loops) effectively remove any 
practical difficulties due to the complexity and 
amount of the analysis involved. The factors which 
prevent the present construction of such a computer 
are not those of computing complexity, speed, etc. 
The real difficulties, and the limitations of such a 
procedure, will be discussed in the next section. But 
it seems wholly likely that these difficulties and 
limitations can, at least for many important prob¬ 
lems, be overcome. And the present author would 
like to risk the prophecy that just as World War I 
saw the birth, and World War II the high develop¬ 
ment and effective use of fire control predictors 


which, so to speak, solve the immediate tactical 
problem for a single gun, so the next war (should 
there be one) may result in the development and use 
of general tactical computers for use in the field (as 
on the flagship of a fleet, for example), and even of 
strategic computers at the disposal of the high com¬ 
mand. 

12.5 CAUTIONS 

The last few pages have described an imaginary 
machine. This description will, it is hoped, serve to 
make clear and vivid certain fundamental aspects of 
what a general theory of air warfare might sometime 
be. But this example, with its “Rube Goldberg” 
machine, runs the risk of giving certain wrong im¬ 
pressions. We want, therefore, to express here some 
warnings. 

1. Will those who consider the possibility of such 
a machine to be wholly controversial or dubious just 
forget the last few pages, and consider the rest of the 
facts and arguments here presented? 

2. No one can hope, at this early stage, to predict 
the actual form of the ultimate science of aerial war¬ 
fare, so details are not to be taken too seriously. 

3. It may very well be that, for a long time in the 
future, it will not be possible to carry out analyses 
of aerial warfare as “overall” in character as those 
indicated in the last few pages. But that would in no 
way reduce the importance of dealing with simpler 
component problems, and then working gradually up 
to the study of whole operations. 

4. The concept of military worth is admittedly, at 
this stage, both vague and difficult. How can one 
measure, with one quantitative index, so many 
tanks and so many soldiers’ lives, and so many dol¬ 
lars, and so many man-hours of labor? I am sure that 
I am not now prepared to answer. It may be that 
more than one index will be unavoidable (and hence 
several output dials on TSC). But there are deep and 
experienced military, economic, and analytical 
minds available to work on this problem, and one 
always returns to the stubborn fact that the overall 
comparison, whether of incommensurables or not, 
has to be made either by analysis, or magic, or blind 
guess. I am simply arguing for facing the complexity 
and the facts, and pushing analysis to its usable 
limit. 

5. Remember that a great deal of progress has 
already been made. Some of these advances are re¬ 
ferred to later in Section 12.6. But apart from the 


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203 


encouraging progress made in the study of military 
problems, both in this war and earlier (as sketched 
in Section 12.6), there has been a great deal of more 
basic mathematical work which is relevant. 

Reference has been made, in Section 12.4, to the 
powerful analysis of competitive processes carried 
out by von Neumann and Morgenstern. 1 There are 
many further important pieces of research to be 
found in the large literature of mathematical eco¬ 
nomics. In addition, the quantitative ecologists, 
aided by outstanding mathematicians, have de¬ 
veloped extensive quantitative theories of the com¬ 
petition of two or more populations for food supplies, 
etc. The literature of competitive biological systems 
has been largely developed by the great Italian 
mathematician, V. Volterra, 2 the American statisti¬ 
cian and biophysicist, A. J. Lotka, 3 the French biolo¬ 
gist, Georges Teissier, and the Russian biologist, 
G. F. Gause. 4 

12.6 DIFFICULTIES, OBJECTIONS, AND 
COUNTER-ARGUMENTS 

In the above descriptive remarks, all sorts of 
difficulties have been overlooked or blithely snow¬ 
plowed ahead; we will pause at this point to take 
some account of these matters. 

To try to argue that a general theory of air war¬ 
fare is easy to construct would be as ridiculous as it 
is unnecessary. The problems are of the gravest pos¬ 
sible importance, and of staggering complexity. But 
the problems are also of compelling character. In 
many instances (and during war, in practically all 
instances) the problems are unlike many questions 
in pure science. The stark fact is that prompt de¬ 
cisions have to be made: a plan — excellent, good, 
mediocre, or disastrous — has to be adopted and 
followed. It is not a question of mere intellectual 
curiosity; there is not an indefinite amount of time 
during which we may philosophize; and it may be 
utterly disastrous to learn only by trial and error. 

Under such circumstances it is really futile and 
irrelevant to keep remarking that the problems are 
complicated, subtle, and difficult. Of course they are! 
But the brutal fact is that they must, for better or 
worse, be solved. The relevant question is: how? 

Is there any advantage to be gained from refusing 
to admit or face complexities and difficulties which 
are in fact inherent? How do “judgment” and “ex¬ 
perience” and “common sense” cope with such prob¬ 
lems? In some instances it must be confessed that 


they do not cope at all, and that these words are 
merely used as labels to cover up a prejudiced guess 
made by someone in authority. In other cases — in 
most cases — there is undoubtedly an eager and 
honest attempt to invoke experience and to exercise 
judgment and common sense. But judgment based 
on experience, to the extent that it is rationally 
utilized, must face the problem of deciding that 
certain features of the present do or do not correlate 
with the past — that is to say, must use the ana¬ 
lytical method. And what, indeed, are such efforts 
other than disorganized and feebly intuitive shadows 
of a real analysis? 

Any one who, with a background of scientific 
training, has watched the procedure of decision as it 
sometimes operates in military circles, has very 
likely ended up with a tremendous personal attrac¬ 
tion for the vast majority of sincere, intelligent, and 
patriotic officers, but with disappointment because 
of the frequent lack of technical competence at 
decision levels, and with distrust for what sometimes 
travels as “common sense.” 

One of the most paralyzing sets of circumstances in 
which to meet the difficulties of unsoundly based 
common sense is in connection with what one may call 
the “hero problem.” A young and extremely capable 
pilot goes through a long and terrible siege of combat 
experience. He is, by this time, a lieutenant colonel 
or even a colonel. He is intelligent, and he obviously 
is experienced. He is brought back to Eglin, or 
Orlando, or Wright Field, or Washington, and he is 
suddenly put in a position of very considerable re¬ 
sponsibility with jurisdiction over technical decisions. 
He may, for example, be concerned with sights and 
sighting mechanisms. An engineer, physicist, or math¬ 
ematician who has been studying sights intensively 
and exclusively for several years comes to his desk 
for information, or for backing. The colonel thumbs 
through the material, turns a little pale at the 
mathematics, and then seizes on some one feature of 
the sight which he thinks is like some other sight he 
once tried and didn’t like. At once his front-line 
vocabulary springs into action, and he makes it 
purply clear that he disapproves. If the technical 
expert says — “But Colonel, don’t you think . . .” 
He is interrupted with, “Say, listen; I have been up 
there with those * * * * * * shooting at me, and I 
know what I am talking about!” 

And you look at the triple row of ribbons on his 
chest, and you are devoutly thankful to him for his 
youth, skill, and bravery — and there is just no 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


decent way in which you, a theoretical stay-at-home, 
can tell him that he is wrong. 

That is what was meant, above, by the “hero prob¬ 
lem/’ The worst of it is that this energetic, patriotic, 
and altogether fine young man is often wrong. There 
are essential complexities and subtleties to the 
problem which he has never thought of, and for 
which his technical training does not at all equip him. 
His own ideas have been inflexibly set by his own ex¬ 
perience under fire. He may have used a very bad 
sight and very bad tactics — but if they, in fact, 
carried him through alive, it is beyond reason or hope 
that he will ever be objective about them. He may 
even be the sort who thinks that the scientists are 
wasting their time interfering with soldiers in what is 
after all their business (as if it were not the business 
of all of us!). And he probably thinks that both for 
automatic weapon antiaircraft and plane-to-plane 
fire, “all you have to do is look at the tracers! Then, 
Doc, you see it all with your own eyes, and you 
don’t need any of the goddam mathematics.” This 
is what he calls “common sense.” 

It must be remembered that individual combat, as 
experienced in the Air Forces, is a highly variable 
experiment in probability, so that these individual 
experiences do not mean much apart from the total 
evidence (some of the most important of which 
would have to be given by dead men). Success is 
often achieved because of the personal characteristics 
of the man, and in spite of rather than because of his 
equipment. The testimony of outstanding individuals 
is therefore chiefly an argument for outstanding 
individuals, rather than for or against any particular 
pieces of equipment or any particular techniques. 

There is no intention to imply that such well- 
intentioned and excellent, but misplaced officers 
constitute the majority. On the contrarjq one of the 
strongest and happiest impressions of the scientists 
who have worked with the Services is the large num¬ 
ber of really excellent officers — highly intelligent 
and energetic, and men of fine character. The point 
being made here is that: 

Officers required to administer technical programs 
and to make recommendations on technical developments 
should he chosen for their technical training , not their 
battle experience. Such officers should he given credit for 
technical service, and should he maintained stably in 
such service long enough to make their technical compe¬ 
tence effective. 

One of the most outstanding characteristics of a 
technically competent man should be a knowledge of 


his own limitations, and a willingness to seek and 
accept more expert advice than he can himself 
furnish. He should be able to bring an unemotional 
logic to bear on the survey of problems, and he should 
have a capacity to recognize and appreciate the 
major points in a technical argument, even though he 
personally could not construct the argument. All of 
these comments are closely connected with the de¬ 
sirability of the briefing of officers at the decision 
level by a competent specialized staff. 

There is good evidence that the points just empha¬ 
sized are well recognized by certain highly placed 
officers — by General Arnold himself, for example. 
But unfortunately he plans to retire, and a new set 
of top brass must be convinced all over again. 

But let us return to our main thesis. We have listed 
above a considerable number of the kinds of factors 
which enter into air warfare problems. 

No one, presumably, would argue that such factors 
are unimportant. And if there are other, dominantly 
more important factors, then by all means let them 
be brought forward by their proponents, and it will 
at once be agreed that these factors (after their im¬ 
portance has indeed been established) should be in¬ 
cluded in such a scheme. 

No one would presumably deny that military oper¬ 
ations should be conducted in accordance with the 
plan which promises to give the greatest net return. 
And who is prepared to claim that, on the basis of 
experience and intuition and judgment, he can 
instantly and mysteriously produce solutions which 
he denies can be produced or improved by logic and 
analysis? The concept of military worth may very 
well be a most difficult one, hard to apply to specific 
cases. But if it is in fact a basic concept which plays 
an inevitable, even though often a hidden and un¬ 
recognized role, what virtue is there in disregarding 
it? Nothing is to be gained by the ostrich-like pro¬ 
cedure of burying our heads in the sand to try to 
escape complication and difficulty. The complica¬ 
tions have to be faced, and the sooner they are 
brought out into the open and subjected to logical, 
quantitative analysis, the better. 

It will be urged by some that it is very difficult, if 
not impossible, to devise any quantitative measures 
of worth Avhich can be applied alike to factories, 
ships, food, aircraft, men, etc. Admittedly and 
assuredly it is difficult, but it is paralyzing and self- 
defeating to admit that it is impossible. In fact, any 
judgment, including those produced on the mysteri¬ 
ous basis of “judgment” or “experience,” actually 


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DIFFICULTIES, OBJECTIONS, AND COUNTER-ARGUMENTS 


205 


and necessarily involves such evaluations. It merely 
involves them — which would not seem to be any 
specially proud advantage — in a concealed and 
unanalyzable way. 

To urge that the problems are too complicated to 
solve is (not to repeat again the obvious rejoinder 
that, complicated or not, they have to be solved) to 
deny the whole history of man’s conquest, through 
the imaginative power of the analytical mind, of 
complicated problems. Suppose that the atomic 
physicists or the cosmologists or the geneticists had 
been content to say, “No, the facts are too subtle 
and inaccessible, the problems too complicated!” 

It is, moreover, important to recognize that useful 
results can come from a start which is admittedly 
incomplete and inaccurate. A precise analysis of 
relationships is sometimes possible even though the 
numerical values of the various parameters be wholly 
unknown. And then it may turn out that certain 
very important conclusions flow, even though one 
has to make wild estimates of certain difficult param¬ 
eters. 

This last point is important enough to deserve il¬ 
lustration. Suppose one wishes to compare the rela¬ 
tive effectiveness of two alternative armaments for a 
fighter plane (say eight .50-caliber wing guns versus 
four such wing guns and two central cannon). This 
comparison depends on a considerable number of 
factors — fire control accuracies and vulnerabilities 
chiefly — which are very difficult to estimate with 
any accuracy. But if it turns out that one asks sev¬ 
eral persons to estimate the fire control accuracies 
and the vulnerabilities, and receives very disparate 
replies, and if, in spite of this disagreement, it turns 
out that for each and every combination of guesses, 
one armament turns out to be superior to the other, 
then clearly the analysis has produced a result that 
overrides the difficulties of estimation of parameters. 0 

Many problems of warfare involve probability 
considerations, and this is a field which is notoriously 
tricky, and within which “common sense” is often 
quite helpless. The author, for example, knows one 
little probability question which can very nicely be 
used as a basis for bets. Actually the odds for the 
person who understands the game are about three 
to one; but it is so deceptive, and seems to any 


c P. M. S. Blackett has said: “No pregnant problem should 
be left unattempted for lack of exact numerical data, for often 
it is found on doing the analysis that some significant con¬ 
clusions recommending concrete action can be drawn even 
with very rough data.” 


common-sense person so fair, that the author has 
felt himself morally restrained from playing the game 
with any persons other than professional statisticians. 

Examples of a similar sort can very readily be 
found in military problems. For example, what is the 
optimum mixture of armor-piercing [AP] and in¬ 
cendiary ammunition for the rear guns of a bomber? 
Specifications often designate such mixtures as five 
AP to two incendiary (we are neglecting tracers here). 
Why? The somewhat striking and by no means obvi¬ 
ous fact is that given any fixed type of target it is 
better to have either all AP or all incendiary, de¬ 
pending on the nature of the target. The justification 
for any other intermediate mixture should be based 
on knowledge of the relative probability of encounter¬ 
ing different targets, certain of which would be more 
vulnerable to AP, and others more vulnerable to 
incendiary. 5 

As another example, consider the effect of a 
weather forecast for a certain future military oper¬ 
ation. To oversimplify the problem, suppose that one 
is only interested in whether the weather is good (G) 
or bad (B). Suppose that one can in some way assign 
the relative military worth (MW) for the four pos¬ 
sible cases that: 

1. The forecast is G and the weather actually 
turns out to be G. 

2. The forecast is G and the weather actually 
turns out to be B. 

3. The forecast is B and the weather actually 
turns out to be B. 

4. The forecast is B and the weather actually 
turns out to be G. 

In a given case, for example, the MWi of case 1 might 
be exceedingly large and positive, and the MW 2 of 
case 2 very large and negative; while MW 3 and MW 4 
might both be moderate or small. (This would be the 
case of an operation which is very important if suc¬ 
cessful, disastrous if unsuccessful, but which could 
be postponed until later without great loss.) What 
should be the weather forecast if one wishes to 
maximize the expected worth of the operation? 

Again the answer is striking, simple, and not ob¬ 
vious. Anyone who is interested can find a discussion 
of this problem in a reference. 651 

To conclude these two examples it may be worth 
while to point out that, from a mathematical point of 
view, they are not separate and different examples, 
but merely two illustrations of a single, simple, basic, 
analytical fact. This point is mentioned because it is 
by no means trivial. One of the important reasons 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


the analytical approach is powerful is that it strips 
off the confusing superficial aspects of problems, and 
exposes their inner logical structure. And often, when 
this inner structure is exposed, it turns out to be 
logically equivalent to some other situation already 
studied and solved. 

In certain cases the analysis will reveal that the 
final result (say the military worth) is extremely in¬ 
sensible to the value of a certain parameter. That 
fact of itself may be of great importance. In other 
cases it may turn out to be impossible (or in any 
event impractical) to assign sufficiently definite 
values to the various basic variables as to permit 
anything like a unique determination of the military 
worth of a given plan. But it may nevertheless be of 
extreme importance to delimit the total range of 
possibilities. This is in a way analogous to the pro¬ 
cedure of the cosmologist. He does not have enough 
nor precise enough information to construct a com¬ 
plete theory of the universe. But he can succeed in 
establishing the fact that, of all possible universes, 
the actual one has properties which lie within certain 
specifiable ranges. And then he has a definite frame¬ 
work of procedure, and can systematically refine his 
analysis and narrow its limits of uncertainty. 

There is, moreover, further impressive and practi¬ 
cal evidence in favor of attempting the quantitative 
analysis of complex situations involving elements 
which seem to elude quantitative specification. For 
this, in fact, is precisely what was done with such 
outstanding success by certain operational research 
groups during the war. Professor P. M. S. Blackett, 
to whom is due so much of the success of the British 
Operational Research activity, has set forth the es¬ 
sence of this matter in a brilliant and deceptively 
obvious, but really profound paper entitled, “A 
Note on Certain Aspects of the Methodology of 
Operational Research.” 7 And for a classic example 
of success in dealing with a very complex situation, 
the reader should study the Admiralty Operational 
Research reports on the convoy problem, or the clas¬ 
sic and critically important paper, “Air Offensive 
Against U-Boats in the Bay of Biscay.” 

It is recognized that practically all of the really 
broad military problems are characterized by great 
complexity and by the lack of a great deal of accurate 
numerical data. The problems are, as has been em¬ 
phasized by Blackett, more like the problems of 
biology and economics d than the problems of physics. 


d Remember the concluding remarks of Section 12.4. 


And therefore one cannot reasonably hope that, at 
least for some time to come, we will have complete 
logical theories of the a priori sort which have been 
so successful in physics or astronomy. One must 
usually be content, as Blackett states, with vari¬ 
ational theories which, like estimates of marginal 
profit, study the response of the military worth to 
changes in the decision variables, often using statisti¬ 
cal as well as analytical methods of estimating 
differential coefficients, and making maximum use 
of operational constants as well as operational func¬ 
tions. 

The operational constants and the operational 
functions constitute some of the most precious and 
dearly-bought results of combat experience. These 
results should be distilled out of the every day ex¬ 
perience at the front by the Operations Research 
personnel, and should be importantly supplemented 
by overall comparisons of the effectiveness of oper¬ 
ations in various theatres. 

This latter type of evaluation has, granted the lack 
of adequate technical and analytical assistance, been 
carried out during World War II with skill and 
objectivity by various groups and for various areas. 
A good example, for instance, is the report of the 
AAF Evaluation Board, Pacific Area . 18 A detailed 
examination of this honest and able report would 
convince anyone that errors had been made which 
could have been anticipated and prevented had suffi¬ 
ciently broad studies, of the sort urged herewith, 
been carried out and the results utilized. 

Throughout this chapter the atmosphere of the dis¬ 
cussion is the atmosphere of war. Many of the re¬ 
marks imply that studies were being made while a war 
is going on. These studies are absolutely essential, for 
the complexity and other characteristics of the prob¬ 
lems are such that progress cannot very well be made 
without knowledge of the operational constants and 
operational functions mentioned just above. 

But even more emphasis should be given to the 
necessity that such subjects be studied during the years 
of peace. Precisely because the problems are so diffi¬ 
cult, we have little chance of making important prog¬ 
ress and of being in a strong and well-prepared posi¬ 
tion unless a substantial and a sustained research 
program in this field be carried out after the war. 

Between wars it is difficult if not impossible to 
give attention effectively to detailed (especially oper¬ 
ational) problems of the sort referred to later in this 
chapter as micro-problems. But just for that reason 
we should, between wars, seek to develop the frame- 


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207 


work and technique for the solution of the broad 
(the macro-) problems. 

12.7 PROGRESS IN WORLD WAR II 

As has been indicated by the remarks of the 
last section, there was substantial progress during 
this war in the quantitative analysis of military 
problems. Indeed, the utilization of technical experts 
is by no means new or recent. The Trojan Horse was 
presumably designed by some ancient Omicron 
Sigma Rho Delta. Archimedes was the V. Bush of 
Syracuse. Napoleon had a board of technical ad¬ 
visers. Those who urge large extension of the use of 
analytical methods will, of course, be confronted by 
two extreme groups: one will be quite certain that 
the whole idea is visionary, unnecessary, and impos¬ 
sible; the other will be equally certain that propa¬ 
ganda for the idea is quite unnecessary because the 
notion is so old and so obvious that everyone has 
accepted it. 

The present author’s association with research and 
development problems in World War I, though inti¬ 
mate, was on a very junior level, so that he is not in 
a position to prove what he nevertheless is reasonably 
sure is true. Namely, that there was at that time very 
little effective use of quantitative analytical thinking 
in connection with the instrument development, 
and probably none in connection with tactical or 
strategic plans. It is, however, interesting to note 
that Frederick William Lanchester wrote, in 1914, a 
series of papers entitled, “Aircraft in Warfare: The 
Dawn of the Fourth Arm.” 8 Paper No. V of this 
series contains the paragraph: 

There are many who will be inclined to cavil at any 
mathematical or semi-mathematical treatment of the present 
subject, on the ground that with so many unknown factors, 
such as the morale or leadership of the men, the accounted 
merits or demerits of the weapons, and the still more un¬ 
known ‘chances of war,’ it is ridiculous to pretend to calculate 
anything. The answer to this is simple: the direct numerical 
comparison of the forces engaging in conflict or available in 
the event of war is almost universal. It Is a factor always 
carefully reckoned with by the various military authorities; 
it is discussed ad nauseam in the Press. Yet such direct 
counting of forces is in itself a tacit acceptance of the ap¬ 
plicability of mathematical principles, but confined to a 
special case. To accept without reserve the mere ‘counting of 
the pieces’ as of value, and to deny the more extended ap¬ 
plication of mathematical theory, is as illogical and unin¬ 
telligent as to accept broadly and indiscriminately the bal¬ 
ance and the weighing-machine as instruments of precision, 
but to decline to permit in the latter case any allowance for 
its known inequality of leverage. 


In this paper Lanchester develops, by mathematical 
reasoning, the principle that: 

The fighting strengths of two sources are equal 
when the squares of the numerical strength multi¬ 
plied by the fighting value of the individual units 
are equal. 

During World War II, and not knowing of this 
earlier work of Lanchester, a mathematician associ¬ 
ated with AMP wrote a paper 9 which reproduced 
and extended the previous results of Lanchester, 
giving estimates of times required and victorious 
forces remaining when one military or naval force 
meets a second. The author also investigated the 
quantitative aspects of division of forces, showing, 
for example, that a first force need be only 71 per 
cent as strong as a second to liquidate the latter 
totally if the former be clever enough to maneuver 
the latter into a position where his forces are tem¬ 
porarily divided in two. Other papers in this field 
were written by a Tufts College group working under 
the auspices of the Special Devices Division of the 
Navy. 

Reference has been made above to the brilliant and 
critically useful analyses carried out during this war 
by the British Antiaircraft, Coastal Command, and 
Admiralty Operations Research groups. Similar, and 
also highly successful, groups operated in connection 
with our forces, the outstanding examples being Re¬ 
search Group M which furnished the central analysis 
for the antisubmarine campaign, and the operational 
groups assigned to the various theatre commands by 
the AAF. This is no place, nor is there any need, to 
rehearse the accomplishments of these groups. Look¬ 
ing at our own effort from the outside, but from the 
vantage point of fairly close relationship from the 
time of the first organization of the British AA group 
onward, the author would, however, risk two recom¬ 
mendations. First: 

That every important arm of the Service develop 
Operational Research groups; and that in addition 
to the absolutely essential groups in the theatres , there 
be one or more central correlating groups. 

The absence of such central study and correlation 
seems to this author to be a striking weakness in the 
AAF setup, and an evident strength in the Navy 
antisubmarine setup. Second, and in spite of the 
outstanding personal qualities of the personnel which 
recruited and organized the AAF groups: 

That the recruitment of Operational Research 
personnel , the organization of the system , and the 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


central direction of it be carried out by technically 
trained personnel, including mathematicians, physi¬ 
cists, engineers, psychologists, physiologists, etc., 
but with a minimum of lawyers and architects. 

In England, in addition to the Operational Re¬ 
search groups, a large amount of general analysis of 
air warfare was carried out by the extensive group 
working under the direction of Dr. L. B. C. Cunning¬ 
ham. Certain of these studies relate to rather specific 
details; but others are grave and useful attempts to 
tackle rather broad problems, such as Cunningham’s 
own extensive analysis of air duels. e 

Many bureaus and divisions of our Army and 
Navy recruited and utilized, often with great effec¬ 
tiveness, both civilian and uniformed analysts. 
Aberdeen Proving Ground (particularly the Ballistic 
Research Laboratory), the Ordnance Department, 
the Naval Bureau of Ordnance and the Bureau of 
Aeronautics (particularly at the Naval Research 
Laboratory, Inyokern, Patuxent, etc.), the Army Air 
Forces (particularly at Eglin, Orlando, Wright Field, 
and Laredo) are only a few scattered but important 
instances with which the author happens to be fa¬ 
miliar. Many divisions of the OSRD had large and 
excellent groups of analysts of various types, Di¬ 
vision 14 (Radar), Division 6 (Subsurface Warfare), 
Division 7 (Fire Control), Section T, and the Ap¬ 
plied Mathematics Panel, being perhaps outstanding 
instances. 

In many — perhaps in almost all — of the in¬ 
stances mentioned in the scattered and fragmentary 
list just given, the analyses were related to what 
might be called the micro-theory of war, rather than 
the macro-theory. These two expressions can be 
given more specific definition after certain discus¬ 
sions of the next and concluding sections of this 
chapter. But by macro-theory is meant the general 
and broad analytical theories discussed in Section 
12.4, whereas by micro-theory is meant the analytical 
study of much more detailed problems which can be 
isolated from general tactical and strategic consider¬ 
ations, such as, for example, a study of how much 
the theoretical accuracy of a bombsight can be im¬ 
proved by certain changes in design. If, in contrast, 
one wishes to study not the theoretical accuracy but 
the practical effectiveness of the modified bombsight, 
then he must take into account all sorts of general 
considerations (training, operational simplicity, lo¬ 
gistics, time delays in changing equipment and 

e See Section 12.7.1. 


techniques, value of increased accuracy, operating 
conditions in theatre, nature of target, etc.); and 
then one is perforce faced with a macro-problem. 

The great progress made during World War II by 
many groups and agencies in the solution of micro¬ 
problems is of real significance for a macro-theory; for 
the structure of the macro-theory requires, so to speak, 
bricks to be laid up into the walls, and general blue¬ 
prints for the design. The individual bricks are, in 
many instances, the solutions of micro-problems. If 
these solutions were not available, and if we had not 
thus learned how to make bricks, then there surely 
would be little sense in trying to draw up elaborate 
blueprints of general plans. 

A very large fraction of the activities of AMP was 
directed toward solutions of military micro-prob¬ 
lems. Various of the more important fields of micro¬ 
problems are reviewed in other portions of AMP’s 
final report and many micro-problems in air war¬ 
fare are referred to in other chapters of this volume. 
It is, however, a specific duty of this chapter to report 
briefly on two studies, carried out under AMP aus¬ 
pices, which, while not actually macro-theoretic in 
character, at least come closer to having general 
tactical or strategic scope than do most other AMP 
studies. 

The former of these studies began, before AMP 
was created, under the auspices of the analytical 
section of the Fire Control Division of the NDRC, 
that is, under Section 5 of Division 7. Since the 
Chief of AMP is also Chief of Section 5, Division 7, 
the later transition to AMP auspices was hardly 
noticeable. The latter study to be reported here 
came rather late in the history of AMP. Thus 
AMP, whatever it did in between, started and more 
or less ended with fairly general studies of air war¬ 
fare. We will now briefly describe these two studies. 

12.7.1 Study of Alternative 

Fighter-Plane Armaments 

This study arose out of the enthusiasm which a 
few of us had for two powerful and pioneering papers 
concerning a mathematical theory of air combat. 1011 
We showed and praised these papers to various 
officers of the Army Air Forces and of the Naval 
Bureau of Aeronautics until, in self defense, they 
suggested that we try to digest and simplify these 
papers, interpreting them in terms not so formidably 
mathematical. Then, when they had been aided to 


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209 


an understanding of what had been done, they pro¬ 
posed that we apply the same or similar methods to 
a problem they would set. 

The first step was accomplished by two reports, 
the first of which 12 gave an outline of the two British 
papers, and the second of which 13 gave a nonmathe- 
matical exposition of the two papers. These reports 
were made the basis for an extended conference with 
AAF, Bureau of Aeronautics, and RAF personnel; 
we were then asked, as the second step, to apply 
these ideas to the problem of comparing the relative 
effectiveness of four 20-mm guns versus eight 0.50- 
in. guns as armament for a fighter making a stern 
attack on a defended twin-engine bomber. 

The methodology of the analysis of this problem 
was rather carefully set down, and a list was made of 
the kinds and pieces of information (or of estimates, 
or of guesses) which would have to be made before 
the answer would be forthcoming. These questions 
(nature of combat, bomber and fighter armament, 
value and variations of accuracies, ammunition, vul¬ 
nerabilities, etc.), were then discussed at very con¬ 
siderable length with the experienced officers. As a 
result estimates were arrived at which every one 
agreed were almost certain to bracket the true values, 
although in many instances the true values were ad¬ 
mittedly unknown. Thus the analysis was based on 
five alternative assumptions of vulnerability, three 
assumptions concerning bomber firepower index 
(which takes into account rate and accuracy of fire) 
and four assumptions concerning fighter firepower 
index. For four of the five assumptions concerning 
vulnerability the eight 0.50-in. were found to have an 
advantage (ranging from 27 percent to 75 percent) 
over the four 20-mm guns. For the fifth vulnerability 
assumption, the 20-mm armament was preferable, 
having about 20 percent advantage. 14 

The study was thus necessarily inconclusive. It 
did, however, make clear just what sort of informa¬ 
tion was necessary to obtain a conclusive answer, 
and it furnished the necessary analytical methods. 
Furthermore the study rather strongly suggested 
the superiority of the 0.50-in. armament especially 
since it was discovered that the use of an optimum 
ammunition mixture increased the superiority of the 
eight 0.50-in. guns for the first four vulnerability as¬ 
sumptions and reduced the advantage of the 20-mm 
guns to 10 percent in the case of the last vulnera¬ 
bility assumption. 

As by-products, this study also issued important 
memos on the subject of optimum ammunition mix¬ 


tures, 515 on the optimum interrelation of aiming 
and dispersion errors. 1617 

12.7.2 B-29 Studies 

The fighter armament study was largely carried 
out in 1942 and 1943, although not formally con¬ 
cluded until 1944. The B-29 studies, however, did not 
originate until mid-1944. The initiation of AAF 
Project AC-92, which resulted in these B-29 studies 
by AMP, was effected by a communication, dated 
June 14, 1944, from Headquarters, AAF, asking that 
the NDRC collaborate with the AAF “in determin¬ 
ing the most effective tactical application of the 
B-29 airplane.” This communication was the direct 
result of a conference held at Orlando, Florida, under 
the auspices of the Army Air Forces Board. The 
conference report, which was later made an official 
part of the AC-92 directive, stated in part: 

1. That no scientifically controlled or scientifically 
evaluated investigation of various defensive forma¬ 
tions of bomber aircraft had ever been conducted by 
the AAF. 

2. That it was of the utmost importance to con¬ 
duct such investigations. 

3. That no scientifically controlled or evaluated 
investigation had ever been made of the B-29 de¬ 
fensive armament. 

4. That no standard central fire control gunnery 
doctrine had been developed, no standard manual of 
B-29 gunnery developed, nor any detailed training 
standards developed for B-29 gunnery. 

The NDRC was therefore requested, as Project 
AC-92, to collaborate with the AAF in planning, 
organizing, carrying out and evaluating an investi¬ 
gation of the most efficient B-29 formations. Account 
was to be taken of: 

1. Self protection of bomber aircraft from fighter 
by bomber’s own gun. 

2. Self-inflicted damage. 

3. Support fire. 

4. Maneuverability and evasive action. 

5. Effectiveness of bombing. 

6. Formation control. 

7. Flak. 

8. Rockets. 

9. Air-to-air bombing. 

10. Fighter escort. 

In addition, Project AC-92 was specifically asked 
to evaluate the defensive armament of the B-29 (a 
task which, in fact, was also required by item (1) just 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


above), to study the ability of the average gunner to 
learn the effective use of the system, and to develop 
a standard procedure and gunnery doctrine for the 
B-29. 

If anyone is surprised and shocked that such a re¬ 
quest needed to be made in June 1944 relative to the 
B-29 airplane, which was then being produced in large 
numbers, there are two replies which should be made 
to him. The first is that it was both surprising and 
shocking. But the second reply is that this airplane 
was, with almost incredible imagination and energy, 
forced through to combat use on a schedule which 
made it difficult if not impossible to follow an orderly, 
or even a rational procedure. And one should also 
remember how magnificently the actual result justi¬ 
fied all the unorthodox means. 

The directive for Project AC-92 referred to one 
specific aircraft (B-29), but was in other respects 
very broad. Extensive flying experiments were ob¬ 
viously going to be necessary, and to carry out these 
essential flight tests, it was specified that the facili¬ 
ties of the 2nd Air Force would be available. General 
U. G. Ent, then in command of the 2nd Air Force, 
had in fact been one of the leaders in initiating and 
planning the project. 

After some considerable negotiation, this project 
was assigned to AMP, which accepted it reluctantly. 
Although we had by that time developed a sizeable 
personnel expert in the analytical aspects of bombing, 
air gunnery, flak analysis, etc., it was clear that this 
project involved a huge experimental program and 
also clear that organizational difficulties were bound 
to occur. For civilian organization would, in fact, be 
trying to do many things (such as operating military 
aircraft from a military field) that could be done 
properly only under military organization, and this 
was bound to lead to confusion if not to trouble. In 
view, however, of the fact that there seemed to be 
no reasonable alternative, AMP agreed to do the 
best it could. From that time until AC-92 was closed 
out, the Chief of AMP spent essentially all his time 
on this one job. 

To meet the objective of Project AC-92, AMP set 
up a broad program which included: 

A. Experimental tests at Alamogordo, New 
Mexico. 

B. Model experiments at Pasadena. 

C. Dynamic testing of the Central Fire Control 
at Austin, Texas (under Division 7, NDRC). 

D. Gun camera tests at Eglin Field, Florida (AAF 
Board, with collaboration of Division 7 and AMP). 


E. Collaboration with General Electric in tests at 
Brownsville, Texas. 

F. Psychological studies (under Applied Psy¬ 
chology Panel). 

G. Analytical studies of special features of the 
problem such as vulnerability, flak, etc. (carried out 
by AMP under other studies reported in Part III of 
this volume). 

H. Broad analytical studies, such as those which 
relate to bombing effectiveness, general “economic” 
theory of bombing, etc. 

As is briefly indicated by this list, items C, D, E, F, 
and G could be taken over by existing AMP facilities 
or by other parts of the NDRC. It was necessary to 
provide for A, B, and H, and to provide general 
direction and correlation for the whole program. 

As to A, the Alamogordo (and Albuquerque) por¬ 
tion of the program, chiefly involving experimental 
tests and their interpretation, was carried out 
through a contract with the University of New 
Mexico and in very active collaboration with the 
Second Air Force. This work will be further described 
below. 

As to B, the optical model experiments and their 
interpretation were carried out through a contract 
with the Mt. Wilson Solar Observatory (Pasadena) 
of the Carnegie Institution of Washington [CIW]. 
This work will also be further described in Section 
12.7.3 

As to H, and the problem of central supervision 
and correlation of so complicated and far-flung a 
program, a contract was made with Princeton Uni¬ 
versity which made available a group including 
physicists and engineers as well as mathematicians, 
which had had long experience with fire control 
problems. 

In addition a Steering Committee was set up, con¬ 
sisting of two representatives of AMP; one of Di¬ 
vision 7 (Fire Control); one from Operations Com¬ 
mitments and Requirements, AAF Headquarters; 
one from the AAF Board; one from the 20th Air 
Force; and one from the 2nd Air Force. A highly ex¬ 
perienced and expert RAF officer also attended all 
Steering Committee meetings, and took an active 
part in the Alamogordo experiments, and a repre¬ 
sentative of the War Department NDRC Liaison 
Office attended most meetings. 

Although the writer, who was responsible for the 
direction of the project as a whole, steadily and stub¬ 
bornly took the position that the directive (with its 
broad inclusiveness and its emphasis on scientific 


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PROGRESS IN WORLD WAR II 


planning and evaluation of experiments) could not 
possibly be met unless all the items A to H inclusive 
progressed simultaneously, it soon developed that 
the Service personnel were preponderantly interested 
in A, eventually almost equally interested in B, 
rather uninterested (with one exception) in C, D, 
and E, only very mildly interested in F and G, and 
totally (and in most cases emphatically) uninter¬ 
ested in H. 

What we could accomplish in this project depended 
essentially on AAF support and assistance, and we 
did, after all, consider it to be our duty to come as 
close to doing what the AAF wanted done as our 
scientific and personal consciences would permit. 
Thus it is natural and in fact inevitable that Project 
AC-92 faded out of the picture when items A and B 
had produced their major contributions. Items D, F, 
and G eventually produced results which are cer¬ 
tainly of large and lasting value. f But item H, which 
should appear as the crowning jewel in this chapter 
on general studies in air warfare, was a total flop. It 
was a flop, this author is convinced, because it never 
was given a chance of being anything else. Some 
persons in the 20th Air Force were interested, in 
principle at least, but they considered that such stud¬ 
ies should be carried out within their own organiza¬ 
tion, say by their own Operations Analysis personnel. 
The rest of the Service personnel, without whose 
complete backing this part of the work could not go 
on at all, had apparently used up all their interest in 
broad studies in the process of asking for them in the 
directive. 

To make one of those sage observations which are 
so easy to make in retrospect, it is quite clear that the 
AAF should never have allowed itself to get into such 
a position that a very few months before the aircraft 
in question was to enter combat on a large scale, they 
would have to ask a civilian organization to assume 
responsibility for planning, conducting, and inter¬ 
preting tests and studies aimed to discover whether 
the defensive armament of this great bomber was 
or was not any good; whether it might be better to 
throw the armament away and trust to altitude and 
speed; how to operate the fire control system; how 
to choose and train personnel; what kind of forma¬ 
tions to fly and how to fly them; etc. 


f As the project developed, one important item was added 
— the problem of devising workable laboratory and field 
methods of harmonizing the B-29 guns. This was worked 
out cooperatively with Section 1 of Division 16, and it is the 
writer’s impression that very important results were achieved. 


211 

y 7 

v 

Such questions obviously should, in any reasonable 
and rational development, have been settled long 
before, by test establishments, analysis groups, and 
decision groups maintained by and within the AAF. 

* * * 

Having now sketched the technical and organiza¬ 
tion background for the B-29 studies, it remains only 
to describe the work itself. 

In the early spring of 1944 a group at the Uni¬ 
versity of New Mexico had collaborated with the 
2nd Air Force in carrying out some studies of the 
K-3 sight in a Sperry upper local turret and of an 
N-6 sight in a Martin upper turret, these being 
camera tracking tests with the turrets mounted on 
movable platforms. Previous studies by the same 
group of the stability of the B-17 in flight indicated 
the types and amounts of movements which should 
be given the platforms. 

These studies stimulated the interest of the New 
Mexico group which, together with General Ent, 
played an important role in the Orlando conference 
mentioned above, which resulted in Project AC-92. 
It was thus natural that AMP turned at once to the 
University of New Mexico, offering them an NDRC 
contract under which they would organize a group 
at Albuquerque (and at the large B-29 field at 
Alamogordo) to carry out experimental tests on the 
B-29 airplane. This New Mexico program, as should 
be clear from the explanation above, actually con¬ 
stituted a large portion of AMP’s activity under 
Directive AC-92. The remainder (except for the 
Pasadena optical studies which will be described in 
Section 12.7.3) consisted primarily of the labors of 
the Princeton group in correlating and otherwise 
assisting 8 the large and widely distributed pattern 
of activities in AC-92 by frequent visits, by continual 
assistance in obtaining necessary equipment and 
information, especially from the AAF, by needling 
the various organizations into action on our work, 
by extending specially valuable aid to the Applied 
Psychology Panel in their part of the program, by 
carrying out analyses of various special problems, 
and by trying to keep all parts of this program 
mutually informed through a series of frequent 
“AC-92 Bulletins.” 19 It had originally been planned 
that the Princeton group would also gather together 
all the results of the separate tests and analytical 
studies, and synthesize them into some more general 

« A great deal of such assistance was also furnished by the 
Technical Aides in the central office of AMP. 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


attack on the original objective of determining the 
most effective tactical use of the B-29 airplane. 
This synthesis, as has been stated above, was not 
attempted for the very simple reason that it could 
not go forward without the complete backing of the 
AAF officials, and, as explained above, it turned out 
that in spite of the wording of the official directive, 
they actually just were not interested. 

A brief description of the New Mexico program 
will now be carried out under the four headings: 
Problem, Facilities, Techniques, Results. 

Problem 

Two main problems were attacked: 

1. In what way, how frequently, and how effec¬ 
tively can fighter planes attack a single h standard 
or stripped B-29? 

This involved a determination, at various speeds 
and altitudes, of the types of possible fighter passes, 
the frequency with which the passes could be carried 
out, and the accuracy and duration of the fighter’s 
fire during these passes. 1 

2. What is the defensive strength of a single B-29 
bomber against fighter attack? 

This involved a determination of the accuracy and 
duration of fire from bombers’ guns. Air tests of these 
accuracies were to be supplemented with air-to- 
ground firings, and ground tests of the accuracy of 
the central fire control system. 

In addition to these two main problems, certain 
other problems were studied, either because they 
were necessary and/or inevitable concomitants of 
the two studies just mentioned, or for the practical 
reason that the New Mexico group developed a 
facility which was able to take on and push through 
to necessarily approximate but remarkably rapid 
conclusion a test of almost any aerial gunnery 
device. Since the AAF did not itself have such a 
facility, it was natural that there was a tendency to 
forget the directive (and indeed to forget the NDRC 
itself), and when any sort of a rush problem arose, 
just telephone the boys down in Albuquerque and 
ask them to do a quick job. 


h The New Mexico experiments actually dealt almost ex¬ 
clusively with the problem of single B-29 bombers. This 
was, in any event, the sensible and almost the necessary way 
to start. Some aspects of the problem of bomber formation 
were studied at Pasadena, and in other auxiliary studies such 
as those reported in Part III of this volume. 

! Such a study should, of course, include all possible types 
of evasive action on the part of the bomber. Time did not 
permit this. 


Among auxiliary problems that were closely con¬ 
nected with the original job should be mentioned an 
extensive amount of time spent in determining the 
performance characteristics of the B-29 airplane at 
different altitudes, speeds, loads, and armor and 
armaments. There were also studies of dispersion 
from the various B-29 guns, harmonization, etc. 

Among the disconnected studies should be men¬ 
tioned : 

1. Test of APG-5 with K-3 sight in B-17. 

2. Speed reduction of B-29 due to radar domes and 
open bomb doors. 

3. Turret dispersion tests, A-26 aircraft. 

4. Performance of APG-15 (radar versus optical 
range). 

5. Test of APG-5 with Mark 18 (K-15) sight. 

6. Test of APG-15 with special GE computer. 

Facilities 

At Albuquerque and Alamogordo, the civilian per¬ 
sonnel on this project included 161 persons, as fol¬ 
lows: 11 major technical persons, 16 research assist¬ 
ants, 14 skilled field assistants, 26 administrative 
assistants, 76 technicians (including a large staff of 
computers), 18 machinists, guards, Service personnel, 
etc. 

At Alamogordo, a special unit of the 2nd Air Force 
(AAF Base Unit 206) was assigned to this project, 
and extremely cordial cooperation was furnished by 
the 2nd Air Force Command. BU 206 included a 
total of approximately 160 Service personnel, of 
which something over 40 were officers. 

Thus taking into account the Princeton group and 
the central AMP office, there were about 350 persons 
involved in all in this study. 

The assigned aircraft totaled 43, as follows: 

8 B-29 16 P-47 

(6 standard, 2 stripped) 6 P-63 

5 B-17 8 Other 

A total of 1,448 separate flights was carried out 
(without accident), including 348 flights of B-29 
craft, 166 of B-17, and 744 of P-47. A total of 138,650 
ft of 16-mm film was exposed in gun cameras. A large 
computing and analysis section was developed at 
Albuquerque. 

Techniques 20-24 

The basic reports must be consulted for details, but 
the main results of this study were obtained by 
carrying out actual flights of bombers and actual 


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213 


attack by fighter planes (with simulation of battle 
conditions for interception, etc.); the results of all 
“firings” of guns of bomber and fighter being re¬ 
corded photographically by gun cameras; this great 
mass of film then being measured and analyzed. The 
gun camera method was essentially a single camera 
technique, making use of distant background 25 
points, such as clouds or mountains, as reference 
points. Thus both the terrain and the climate of 
southern New Mexico were essential to the procedure. 
The method is admittedly not as accurate, partic¬ 
ularly for flexible guns, as deflectometer methods, 
but probably sufficed for the purposes in hand. In 
any event, it was several cuts better than nothing, 
and no other known method was really possible, 
taking into account the mass of data which was 
necessary, and the time available. 

A great many other methods were skillfully and 
energetically developed by the New Mexico group 
to handle other details of the program. 

Results 

In this program was experimentally demonstrated, 
and for the first time, as far as we know, the ability 
of the stripped B-29 to carry great loads over great 
distances at great speeds and high altitudes. In 
particular, this project carried out successful ad¬ 
vance simulations of the Saipan-Tokyo and return 
bombing missions. 

The character of the fighter plane attacks which 
can be carried out against a B-29 was experimentally 
determined, for altitudes ranging from 20,000 ft to 
32,000 ft, and the errors of various types of comput¬ 
ing sights determined for these actual courses. 

The accuracy of fighter plane fire against the B-29 
was experimentally determined for altitudes ranging 
from 14,000 ft to 32,000 ft, and the frequency of hits 
analyzed for different approaches, etc. These tests 
were all made with fighter pilots who had had combat 
experience. 

Fighter hits on bomber per pass were obtained as 
a function of angle and range, and converted through 
a study of pass frequency to fighter hits per hour as 
a function of angle and range. 

Gun and turret dispersions were measured for the 
B-29, as well as harmonization errors on the ground 
and (approximately) in flight. Tracking errors were 
determined carefully for all positions, and actual gun 
errors determined approximately. 

Certain inherent limitations of the B-29 central 
fire control system were determined in ground tests 


which measured the output of the whole fire control 
system for known and controlled inputs. 

The ability of the P-47 fighter and of certain faster 
and more modern fighters to attack a stripped B-29, 
flying at high speed and great altitude, was deter¬ 
mined. 

All these jobs could be done better now, or could 
have been done better then if more time had been 
available. But it should be mentioned that the ex¬ 
perimental program did not begin until early July 
1944, and the main results were reported to the AAF 
in accordance with the requested schedule, 20 namely, 
on November 15, 1944, less than five months later. 

It also seems almost unnecessary to add that this 
study demonstrated that, in a reasonable length of 
time, it is possible to obtain the kinds of experimental 
data necessary for a more general, more inclusive, 
and more penetrating study of the employment of 
large bombers. 

12.7.3 Optical Studies at Pasadena 

Just as the experimental program at New Mexico 
was largely based on the previous experience and 
accumulated enthusiasm of the New Mexico group, 
so the model experiment program at Pasadena was 
primarily based on the previous experience and ac¬ 
cumulated enthusiasm of a member of the staff of 
the war research division of the Mt. Wilson Solar 
Observatory, who had had special experience with 
many sorts of optical devices and procedures, and 
who had more recently become well informed con¬ 
cerning aerial gunnery. Shortly after Project AC-92 
was accepted by AMP, a contract with the Carnegie 
Institution of Washington was recommended, under 
which was made available the very unusual facilities, 
both of personnel and of shop equipment and ex¬ 
perience, of the Mt. Wilson Solar Observatory of the 
CIW at Pasadena, California. 

We will now briefly describe these Pasadena studies 
under the four headings: Problem, Facilities, Tech¬ 
niques, Results. 

Problem 

To analyze the intensity and distribution of fire¬ 
power about variously sized and designed squadrons 
of (B-29) aircraft, and 

To furnish a method whereby these complex re¬ 
lationships can be visualized. 

The complexity of the situation can be appreciated 
when one notes that a single B-29 has 5 turrets and 


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COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


5 sighting stations, with 36 independent arrange¬ 
ments of the controls of the various turrets by the 
various sighting stations. 

Facilities 

Employed under the contract were four senior 
scientific men and six assistants. In addition, Mt. 
Wilson Observatory donated the full or part-time 
services of 19 of their own staff; so that nearly 30 
persons were involved in toto. The program was 
carried out under a local Directive Council headed by 
the Director of the Mt. Wilson Observatory. 

Shop and other facilities (particularly computa¬ 
tional) were made available by the Observatory, and 
the gymnasium of a local school was rented in order 
to have a room large and high enough to accommo¬ 
date the experimental setup. 

Techniques 26-28 

This study was based upon the idea of making 
small (actually 1 to 72 scale) model planes, and in¬ 
stalling a small light source at the position of each 
gun (and of each sighting station). These lights were 
equipped with shields which restricted the field of 
illumination in accordance with whatever restrictions 
there might be on the field of fire of the gun in ques¬ 
tion (or the field of visibility of the sighting station 
in question). A switching system permitted the 
simulation of all the possible methods of control of 
the various guns by the various sights. Then since 
the effectiveness of fire varies roughly with the in¬ 
verse square of distance, as does the intensity of 
illumination from a point source, the total intensity 
of illumination of any point in space was approxi¬ 
mately proportional to the total firepower which can 
be brought to bear at this point. 

Thus one constructs a set of models, installs the 
lights and shields, adjusts individual intensities in a 
calibrating operation, arranges the models in space 
to conform to some formation design to be studied, 
and then determines the intensity and distribution 
of firepower for various briefing systems by measur¬ 
ing the corresponding light intensity pattern. 

The model squadrons were set up at the center of 
a 25-ft radius spherical shell, on the white inside 
surface of which one could actually observe the light 
intensity pattern. Some 18 model planes were con¬ 
structed for use at Pasadena, and 12 more for use in 
the Marianas. 

The light intensities could be measured, point by 
point, with photocells. This method was developed 


to an accuracy of about 1 per cent. Or one could 
photographically record the light intensity pattern 
over some area, and then carry out photometric 
measurements. Or, still more simply, one could 
simply count the lights visible from a certain point, 
i.e., the guns which could be brought to bear on this 
point, tape distances, and rapidly compute fire- 
powers. Although this last method does not make 
full use of the optical scheme (and permits, for 
example, any law of decay of firepower with dis¬ 
tance), it actually proved to be the easiest and quick¬ 
est to use in many instances. 

Results 

Using the system described above, an exhaustive 
but preliminary study was made of the firepower 
pattern for a single B-29 with all of the various types 
of control. 29 Then similar studies were made for a 
squadron of 4 planes, 30-32 for the 11-plane squadron 
as used by the 21st Bomber Command, 33 and for 
the 12-plane squadron as used by the 20th Bomber 
Command. 34 

In February 1945, the 20th Air Force held a large 
conference 35 at Pasadena, primarily to observe and 
consider the results thus far obtained in this study. 
As a result of this conference certain changes were 
recommended in the stacking and briefing of the 
12-plane squadron; and this modified 12-plane squad¬ 
ron was then thoroughly analyzed by the Mt. Wilson 
group. 36 

Early in this project it became obvious that one of 
its greatest contributions was that it furnished a 
method whereby one could see , simply and con¬ 
vincingly, a pattern of relationships otherwise so 
complicated and subtle as to provide topics for per¬ 
petual arguments. Two officers, urging different pat¬ 
terns for a formation, could set them up on this 
optical model, and could together directly examine 
the resulting pattern of firepower. 

Thus it was clear that the device had important 
applications for training, and in the field as well as in 
a scientific laboratory. Moving pictures were made, 
showing the firepower variation of formations as one 
circles about them. Concerning such pictures the 
President of the Army Air Forces Board remarked 
that he “believed these motion pictures gave the best 
idea to airmen as to the relative effect of firepower 
about a formation yet presented.” Certain of these 
pictures were flown to the Marianas and viewed 
by General LeMay and by many gunnery officers and 
men at the front. And as was indicated above, a 


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MATHEMATICAL ANALYSES AND WAR 


215 


model setup was constructed and flown to Guam for 
use there in analyzing firepower problems. 

In addition to the successful achievement of their 
primary aim, the personnel of this project carried 
out and reported on certain auxiliary studies. A con¬ 
siderable analysis, for example, was made of fighter 
attacks, 37-39 and it was shown that for a wide va¬ 
riety of attacks, all the breakaways pass through a 
limited region some 20° in angular width and some 
400 yd off the stern. Studies were also made of the 
problem of offset guns in fighters, 40 and of the 
duration of strafing attacks. 

Again the author of this chapter cannot resist em¬ 
phasizing that these studies, although much narrower 
in scope than the “general” studies here being advo¬ 
cated, are excellent examples to show that the in¬ 
formation necessary for the general studies can in 
fact be obtained. 

12.8 MATHEMATICAL ANALYSES AND 
WAR 

The topic of this chapter is “General Theory of Air 
Warfare.” But it has been a central purpose of this 
chapter to point out that modern war is essentially 
and inevitably a combined operation. Thus it seems 
not inappropriate to close with some general remarks 
concerning the contribution that mathematics can 
make to the broad field of national defense. These 
remarks will involve some topics which have not 
been mentioned before in this chapter, and should 
serve to furnish a setting for the general activities 
which are the special concern of this chapter. 

To give any adequately complete discussion of the 
possible contributions of mathematics to w T ar and to 
national defense would require far more space than 
is available here, and quite other authorship. All that 
is intended here is to give that minimum description 
which will serve to indicate the relationship of the 
general analytical studies of warfare, here urged as 
vital, to other mathematical activities of military 
significance. 

We will discuss mathematical activities of military 
importance under the following headings: Basic, 
Service, Design and Use of Individual Devices, Op¬ 
erational Analysis, General Theories. 

This list of headings is poor for several reasons. The 
subheadings are neither exclusive nor self-explana¬ 
tory, and in some cases they are worse than vague, in 
that they appear to say something less than or more 
than is intended. But since some effort has failed to 


produce a better list, let us take these topics, one at 
a time, and explain briefly what is meant. 

12.8.1 Basic 

The most important point is that all developments 
of mathematics, from the most abstract concept of 
the purest of pure mathematics to an isolated item of 
new technique in applied mathematics, are basic con¬ 
tributions to national technical strength. Engineer¬ 
ing, physics, chemistry — yes and biology and medi¬ 
cine as well — use for their central core of quantita¬ 
tive precise thinking the analytical tools provided for 
them by the mathematician. In these days of the 
atomic bomb, one hardly needs to emphasize the 
close relationship between military might in the field, 
and the abstract pencil theories of the analytical ex¬ 
pert. Thus the stimulation and support of the field as 
a whole, especially through fellowship and other train¬ 
ing aids, is both an essential and a sure way to provide 
the mathematical talent required in time of emer¬ 
gency, and is also an essential basis to a proper de¬ 
velopment of other elements of the technical strength 
of the nation. 

Volume 1 of the Summary Technical Report of the 
Applied Mathematics Panel, together with Volume 
3 which is devoted to probability and statistics, 
although primarily concerned with describing some 
of the intermediary steps in the military utilization 
of basic mathematics, does thereby indicate some of 
the classical fields of pure and applied mathematics 
that connect closely and obviously with military 
problems. It is intended to include here, under the 
heading of “Basic,” all such activity. But it is also 
intended — and this is the main point — to include 
here the whole of mathematics as a vitally necessary 
element of scientific strength, basic to all develop¬ 
ments of the other sciences and technologies, and 
hence basic to the national welfare and the national 
defense. 

12.8.2 Service 

This title is an egregious misnomer, for from one 
valid point of view it covers all of mathematics in 
direct relation to military problems. The mathe¬ 
matician who is trying to make his discipline serve 
the national need should, in one sense, have nothing 
hut service in mind. 

But one can serve in many ways. Sometimes one 
can serve best precisely by not doing what the other 


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216 


COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


fellow wants; or by doing something that he has 
never, and would never, think of. And the word 
service is here used in the restricted and limiting sense 
of “doing what the other fellow wants.” We should 
not lose sight of the fact, however, that in the process 
of so doing, one’s major opportunity often lies in 
gently leading the other fellow to see what he ought 
to want. 

There is a great deal of useful mathematical service 
that can be done on the level of isolated requests. 
Thus individuals, or military agencies, or nonmilitary 
or government agencies, or NDRC groups, or uni¬ 
versity and commercial laboratories working on 
national problems, may come and say: 

Please evaluate this integral. 

Please solve this differential equation. 

Please compute the value of this determinant. 

Please interpolate, using third order differences, 
nine values between every two values of this table. 

These requests may arise out of secret projects, and 
it may therefore be necessary or wise (which, by the 
way, are not equivalent) to keep the mathematician 
completely in the dark as to the nature of the prob¬ 
lem. In such cases, these are items of pure service , so 
to speak, where the mathematician (provided he can) 
does precisely what he is asked, and no more. 

It is easy to see, however, that isolated requests 
are extremely likely to lead to another and more 
important grade of service. For if the problem is, 
for example, not to evaluate a single determinant, 
but rather to evaluate many determinants, then the 
mathematician may well inquire as to the original 
formulation of the problem. And he may be able 
to suggest alternative equivalent procedures which 
lead to simpler expressions, easier to compute. Or he 
may refer the client to devices which have been 
constructed for the solution of systems of linear 
algebraic equations. Or he may even uncover errors 
in the formulation of the problem which make it 
undesirable to do any computing at all. 

As another example, suppose the specific question 
to be: Study these two records of firing trials; and 
tell us which gun and fire control system is better, by 
how much, and how certainly. 

Here there are almost surely two stages of services 
involved. First (and it is unwise to omit or delay this 
stage), the service mathematician should accept the 
situation, however bad it is, and do the utmost that 
can be done to answer the questions asked. Second, 
he should point out to the client why the problem 
was a bad one, how the trial could have been im¬ 


proved. This may well open up, to the client, the 
whole field of statistical design of experiments, and 
the second stage of service may well be of indefinitely 
greater significance than the answer to the original 
problem. j 

A great many groups and agencies and establish¬ 
ments, recognizing the value of mathematical service 
of this sort, obtained for themselves what the German 
physicists used to call “house mathematicians,” i.e., 
locally available service mathematicians. The AMP, 
in addition to the service which it was prepared to 
furnish through the personnel of the contractual 
groups in New York, Princeton, Providence, Chicago, 
Cambridge, etc., also farmed out individual mathe¬ 
maticians to various groups on either short- or long¬ 
term loans. One of the conspicuously successful of 
these ventures involved the placement of an AMP 
mathematician with the Army Air Forces Board at 
Orlando, where he served for over twenty months as 
a mathematical consultant. As a result of this ex¬ 
perience, he has written a set of comments and 
recommendations which seem so important that they 
are included, at the end of this chapter, as Appendix 
A. Since the views of this consultant are here repro¬ 
duced, it seems relevant to add that his services were 
so appreciated that he was given, by Brigadier 
General E. L. Eubank, President of the AAF Board, 
a special letter of commendation: 

“. . . for his splendid record of mathematical as¬ 
sistance to the Army Air Forces Board ... he 
has been actively engaged in various projects the 
successful completion of which was, in no small 
part, the direct result of his efforts ... in each 
of the following ... he was a major contributor 
. . . the . . . selection was made since each of 
the projects was widely recognized as having a 
measurable effect upon the war effort and its suc¬ 
cessful termination.” 

12.8.3 Design and Use of Individual 
Devices 

A large fraction of the mathematical work in this 
war has related to the design and use of individual 
devices — antiaircraft gun directors, tracking mech¬ 
anisms, bombsights, proximity fuzes, radar sets, 
plane-to-plane gunsights and sighting systems, rocket 
sights, guided missiles, torpedo sights, etc. In con- 

j This is essentially the way in which AMP got into the field 
of sequential analysis as applied to acceptance testing, quality 
control, etc. 


CONFIDENTIAL 




MATHEMATICAL ANALYSES AND WAR 


217 


nection with such activities there are several very 
important types of questions to consider, such as: 

1. What performance is this device capable of 
when operated perfectly? 

2. What performance is this device capable of 
when operated by a random untrained GI, by a se¬ 
lected untrained GI, by a selected trained GI? 

3. What is the optimum selection and training 
process? (This involves close collaboration with psy¬ 
chologists and physiologists.) 

4. What modifications in design would improve 
(whether they improve or impair the theoretically 
possible performance ) the actual performance with the 
tvpe of operating personnel likely to be available 
in the field? 

5. What is this device supposed to accomplish, and 
to what extent does it do so (that is to say, an analy¬ 
sis of the problem the device is supposed to meet, rather 
than of the device )? 

These particular questions are intended to refer to 
a device which is not available for actual test. They 
involve studies which should be made in time to 
influence design. k The first of the above questions re¬ 
fers, in a fairly strict sense, to what has previously 
been called a micro-problem. The character of the re¬ 
quest is such that an analyst could be handed essen¬ 
tially nothing more than a complete description of 
the design, and could be locked in an otherwise empty 
room until the results were completed. By the time 
one gets down to the fifth question, however, it is 
clear that he has left behind the field of micro-prob¬ 
lems, where one can study one isolated problem 
without reference to a whole large set of practical 
considerations, and has entered the field of macro¬ 
problems. The fourth and fifth questions are bound 
to involve tactical considerations, and may involve 
strategic considerations. 

If one supposes that operable examples of the de¬ 
vice exist, then a new set of problems enter — those 
of testing. One should consider bench testing, both of 
components and of whole devices; use testing under 
controlled conditions which to a lesser or greater ex¬ 
tent attempt to reproduce actual field or combat 
conditions; extended field trials, which are carried out 
on a small scale but under field conditions; and 
finally combat trials. 

One of the great advances of this war, at least in 
those fields with which the author is acquainted, con- 

k This sounds ridiculously obvious, but as a matter of ex¬ 
perience, this condition was by no means always met in the 
cases with which we had experience. 


sisted of eventual recognition by military authorities 
of the necessity for quantitative testing under controlled 
conditions, together with analytical (often statistical) 
assistance in the planning and the interpretation of the 
tests. 

The author was present at some of the demonstra¬ 
tions of military equipment given in the past (about 
four or five years ago). These tests were wholly in¬ 
adequate from the standpoint of controlled scientific 
testing, and certainly should not have constituted a 
basis for acceptance or rejection. 

The advances in objective, quantitative, and con¬ 
trolled testing of fire control equipment, for example, 
have been great and, it is hoped, irreversible. The de¬ 
velopment of dynamic testers and associated comput¬ 
ers for antiaircraft artillery equipment, and the devel¬ 
opment of great testing engines for plane-to-plane fire 
control equipment constitute two great contributions 
to this field, both carried out by Division 7 of the 
NDRC and doubtless fully covered in their reports. 
The development of camera testing procedures and 
the design and assessment of firing test procedures for 
flexible aircraft gunnery, the analysis of frangible 
bullet trials, and a large number of other studies and 
developments were aided by AMP, and are covered 
in Part I of this volume. 1 

One of the important practical results of such 
analysis of device performance, such studies of de¬ 
vice purpose, and such testing of devices as is here 
envisaged is the emergence of some rational basis 
for setting military requirements and also for setting 
the actual production specifications which must be 
met by the manufacturer. Too often some military 
board has in the past put out, with presumably 
straight faces, a set of military requirements which 
come ludicrously close to demanding a device of 
perfect accuracy, simplicity, and reliability, small in 
dimensions and light in weight, demanding no skill 
from the operator, and being producible out of non- 
critical materials in very little time. Too often the 
tolerances set in the manufacturing specifications 
seem to be based not so much on any rational require¬ 
ment, but rather on the notion that the manufacturer 
is having too easy a time of it if any reasonable frac¬ 
tion of his output passes. 

All these questions of device analysis and device 
testing, military requirements, field trials, etc., are 
interrelated in such a way that sensible progress can¬ 
not be made unless the organizational structure of 

1 See also the comments of the previous section of the 
present chapter. 


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218 


COMMENTS ON A GENERAL THEORY OF AIR WARFARE 


the military group involved is such as to permit the 
requisite interlocking of activities. At the request of 
Dr. Edward L. Bowles, Expert Consultant to the 
Secretary of War and Special Consultant to the 
Commanding General, Army Air Forces, the author 
prepared a memo concerning the development of 
assessment facilities for air gunnery in the Army 
Air Forces. Because of its direct bearing on the issues 
just mentioned, this document is reproduced as 
Appendix B of this chapter. It is gratifying to be able 
to add that when Dr. Bowles issued, on April 12, 
1945, a letter of instruction to the man who had just 
been given an advisory appointment in the Secretary 
of War’s office, with special responsibilities in the 
field of aerial gunnery, he outlined the War Depart¬ 
ment’s plans for developing an assessment agency, 
and concluded this section of his letter with the 
direction that the remainder of the pattern was to 
be completed along substantially the lines set forth 
by the memorandum which had been furnished him 
by AMP and which forms Appendix B of this 
chapter. 

12.8.4 Operational Analysis 

This type of activity has been referred to previously 
in this chapter, and it lies outside both the purpose 
and the competence of the present author to add here 
any detailed description or critique of operational 
analysis. It must, however, be included as a major 
item in the list of important ways in which mathe¬ 
matics serves the war effort. 

Operational Research Sections, working in the 
various theatres of war, are concerned with such 
questions as: 

1. How well are we using a certain device? 

2. How can we use it more effectively? 

3. What minor (including field) modification of 
this device would increase its effectiveness? 

4. What devices do we need that we do not have? 

5. What are the causes of our successes and fail¬ 
ures? 

6. What results are we getting against the enemy? 

7. What should we do on the next mission? For 
example: 

a. If we attack on the deck will our wings slice 
through barrage balloon cables without 
amputation? 

b. If we meet jet fighters, what rule of thumb 
should our waist gunners use? 

c. How many bombs should we drop to 


achieve a probability of 0.95 of destroying 
a target? 

It is clear that the success of such groups depends 
essentially on: (1) the personal characteristics of the 
members, particularly their ability to meet field con¬ 
ditions, their attitude of unselfish service, their 
willingness to tolerate inevitable limitations, and 
their capacity to get along with military command 
and with operating personnel; (2) the spread and 
excellence of the scientific techniques which they can 
make available (first-rate pure and applied mathe¬ 
maticians, statisticians, physicists, engineers, psy¬ 
chologists, physiologists, etc.); and (3) the outstand¬ 
ing excellence of their technical leadership. Further¬ 
more, they cannot work to anything like full advan¬ 
tage unless they are backed up by a central home 
agency which correlates and compares their various 
findings, points out duplications of effort and weak 
points in the program of certain groups, and carries 
out those more basic and long-term studies which 
cannot possibly be handled by field groups. 

12.8.5 General Theories 

This topic like the preceding one has been included 
here primarily so that the present list be complete. 
Since this chapter has been chiefly concerned with 
indicating the nature and scope of broad theories 
of air warfare, or of warfare in general, it does not 
seem appropriate or necessary to repeat any of the 
description or argument here. 

It is however hoped that the more detailed com¬ 
ments of the present section on “Mathematical 
Analyses and War” will serve to indicate that the 
general studies — the general plan for the structure 
of defense and war — have available as necessary 
building bricks a vast number of dependable and 
essential individual or micro-studies. 

12.8.6 Additional Comments 

Although we have now concluded the remarks fore¬ 
cast in the outline at the beginning of this section, 
there is one topic which merits some final comments. 
Every field of mathematics, including some of the 
most abstract and erudite, has important contribu¬ 
tions to make. It would seem unnecessary and per¬ 
haps even unfortunate to single out some special 
field for special comment. But the fact remains that 
a great many mathematicians — especially some 
so-called pure mathematicians — have in the past 


CONFIDENTIAL 



MATHEMATICAL ANALYSES AND WAR 


219 


been relatively uninterested in the subject of mathe¬ 
matical statistics, and the fact also remains that 
mathematical statistics has proved to be an exceed¬ 
ingly powerful and useful tool in connection with 
many war problems. Of the three volumes of final 
reports of AMP, one is devoted directly and exclu¬ 
sively to statistics, while the present volume leans 
on that subject heavily. If AMP had done nothing at 
all except what it did in applying mathematical 
statistics, its budget (to put the matter on a very 
practical basis) would have been justified many 
times. One particular application out of very many, 
for example, would have effected money savings, if 
the war had continued until the end of 1945, at a 
rate in excess of one million dollars a month, this 
money saving being less important than the fact 
that the improved method could produce, at the 
fighting fronts, more, and certainly a more accept¬ 
able quality of ammunition. 41 

* * * 

There is one further postscript remark. One has 
known of instances in which our military leaders 
were skeptical or even contemptuous of the impor¬ 
tance of certain weapons or techniques — until we 
learned something about them the hard way, from 
the enemy. And then we suddenly accept the wisdom 
or necessity of these weapons or techniques, and get 
feverishly to work on them. 

It may therefore be desirable to point out that 
when, after the war, the leading aeroballistician of 
Germany was interrogated, he remarked that he 
considered it as good as certain that, by statistical 
study, efficiency factors for battles between forma¬ 


tions can be devised which permit one to measure the 
worth of the armament and of the fire control. This 
same German expert, after describing their studies 
of vulnerability, pursuit and nonpursuit attacks of 
fighters on bombers, combat film analysis, etc., 
remarked that the basic purposes were to analyze air 
warfare and to lay objective foundations for theoretical 
studies of tactico-technical questions. 

To indicate what almost incredible delays are 
sometimes involved before an essential idea is ab¬ 
sorbed into military practice, this enemy expert 
mentioned, concerning the use of gun cameras to 
obtain necessary quantitative data on aerial gunnery, 
that in Germany: 

1. The idea was proposed during the Spanish War, 
1936/7, but turned down by the Service. 

2. The first film strip of a Spitfire kill was assessed 
in November 1940. 

3. The first lecture on this subject was given be¬ 
fore the General Staff of the GAF seven months 
later (June 19, 1941). 

4. The first analysis section was set up nearly ten 
months later (April 1, 1943). 

5. Gunnery assessment was “recognized” by mili¬ 
tary units sometime in 1944. 

6. A central efficient assessment bureau was finally 
set up about seven years (October 1, 1944) after the 
idea was first proposed. 

The present author is glad to be able to make this 
point without the necessity of using local data. In¬ 
deed our record, in the decadent and nonmilitary 
democracy, is a good deal better. But if we want to 
exclude a completely disastrous “next time,” it must 
be much better still. 


CONFIDENTIAL 







APPENDIX A 


General comments of Dr. S. S. Cairns based on his ex¬ 
periences as mathematical consultant to the Army Air 

Forces Board, Orlando. 

The following comments are based on over twenty 
months’ experience with the Army Air Forces Board, 
Orlando, Florida, as a mathematical consultant. The 
writer served in this capacity from February 1944, through 
October 1945, as a member of the Statistical Research 
Group — Princeton, under direction of the Applied Mathe¬ 
matics Panel of the NDRC. These comments are offered 
in the hope that they may prove useful in connection with 
any future plans for similar consulting services, especially 
on the part of military organizations, during either peace 
or war. 

A great need existed at the Board for the services of a 
general mathematical consultant. This need, which will 
presumably continue during peace time, arises from the 
fact that the Board is charged with testing and reporting 
on new equipment, methods of employment, and tactical 
questions, with a view to developing official Army Air 
Forces policies. These duties of the Board call for a com¬ 
bination of personnel having combat experience with those 
having sufficient scientific training. During wartime, up- 
to-date combat experience can be provided by a system of 
assignments of officers fresh from overseas duties. This 
was adequately accomplished at the Board, though the 
officers assigned there were not systematically selected on 
the basis of any peculiar fitness for the work of the Board. 
The empirical knowledge provided by active military serv¬ 
ice needs to be supplemented by the abilities of experi¬ 
enced, scientifically trained personnel. There was a serious 
shortage of such personnel at the Board. It may also be 
noted that there was no widespread appreciation of this 
lack or of the potentialities of analytic methods applied 
to many of the Board’s problems. 

The writer’s position as consultant was somewhat in¬ 
formally established. In the future, it would be well if, in 
the formal statements defining the Board (or any similar 
agency), specific provision were made for a mathematical 
consultant, with a definition of his duties, his privileges, 
and his status. The following suggestions are based on 
the writer’s experiences in Orlando, where these questions 
were settled, more or less satisfactorily, by an evolutionary 
process. Many of the suggestions are afterthoughts. As 
such, they are not to be interpreted as adversely critical of 
anyone concerned. In all probability, favorable action 
would have been taken, whenever practicable, on any of 
these suggestions, had they been offered during the period 
of hostilities: 

1. It is desirable for a consultant to be a civilian, so that 
he may be free from the irrelevant aspects of military 
protocol. As a civilian, he can freely associate with and 
express himself to officers and enlisted men of all ranks and 
grades. Conversely, he can be approached with equal free¬ 


dom by all categories of military personnel. He should, of 
course, recognize the authority of the commanding officer 
of the post to which he is assigned. 

2. The privileges of a consultant to the Board should 
include the opportunity to comment on all official projects, 
not merely on those regarding which he is formally ap¬ 
proached by the project officer. Too frequently, a project 
officer does not realize his own need for technical assistance 
or else does not appreciate the potential value of such help. 
Many Board projects were carried out, both before and 
during the writer’s stay at the Board, which could have 
been improved with the aid of mathematical assistance. 
In the case of some projects, advice was sought too late, 
after tests had been designed and performed in a manner 
inappropriate to the required analysis. Consultation on 
design of tests is of especial importance. 

3. The preceding item implies that the Board consultant 
should participate in the staff meetings at which projects 
are discussed. He should be expected to submit suggestions 
for new projects when they seem appropriate. The writer 
attended a few staff meetings, but only when his presence 
was especially requested in connection with a particular 
project on which he was officially giving assistance. He 
was always free to offer suggestions of any sort through 
any officers on the Board or directly to the President of 
the Board. 

4. Some reliable system should be set up whereby all 
circulating documents of possible interest to the consultant 
should cross his desk. The responsibility of filling out rout¬ 
ing slips at the Board frequently devolved on persons with 
strangely limited notions of the breadth of the consultant’s 
legitimate interests, so that many documents of importance 
to him either never reached him or else came to his atten¬ 
tion only by accident. 

5. The value of a consultant to the Board would be 
enhanced by the services of an assistant capable of dealing 
with routine matters involving straightforward computa¬ 
tions, questions of trigonometry, and so on. 

6. A consultant requires (and the writer received) every 
consideration in the matter of secretarial services, office 
supplies, official mailing privileges, and everything else of 
the sort necessary to the performance of his duties. 

7. A technical consultant should have an office of his 
own, appropriately labeled. The writer came to be as¬ 
sociated, in the minds of most officers at the Board, with 
the Armament Division, so that members of other divisions 
occasionally felt they were borrowing his services when 
they consulted him. This situation arose partly because 
the writer shared an office with members of the Armament 
Division, and partly because Colonel V. C. Huffsmith, 
Chief of that division, was, of all officers on the Board, the 
quickest to appreciate and avail himself of the services of 
the consultant. This situation had certain advantages, 
since the consultant was able to secure Colonel Huff- 
smith’s advice and backing in assigning his own personal 


CONFIDENTIAL 


221 


222 


APPENDIX A 


priorities to different tasks — a very ticklish problem when 
several projects were simultaneously calling for urgent at¬ 
tention. If the recommendations implied in this memo¬ 
randum were followed, a mathematical consultant to the 
Board would be greatly overburdened during wartime, 
unless he was in a position (as was the writer) to refer 
problems promptly to such agencies as the various AMP 
groups. 

8. It should be understood in advance that a technical 
consultant is entitled to military orders when traveling on 
the business of the agency to which he is assigned. The 
writer succeeded in making arrangements for the regular 
use of such orders, authorizing travel by government air¬ 
craft and granting priorities on commercial aircraft, after 
he had served about a year in his post at the Board. 

9. In general, a definite arrangement should be made, 


perhaps by an AGO pass, or other suitable official docu¬ 
ment, whereby a consultant would be assured, wherever 
he went, of all appropriate rights and privileges of a field 
grade officer. These affect assignments to sleeping quarters, 
Officers’ Club privileges, eating in officers’ messes and so on. 
The writer received such considerations, after a reasonable 
period of “breaking the ice,” but they should be recognized 
as prerogatives, rather than as personal courtesies. 

10. During wartime, a consultant to the Board could 
occasionally benefit by visits to active theaters. Such visits 
would almost necessarily entail the availability of more 
than one consultant, so that duties would not suffer during 
prolonged trips. There were occasions when the writer’s 
work required his presence in Orlando, even though cer¬ 
tain experiences elsewhere would probably have increased 
his value to the Board. 


CONFIDENTIAL 



APPENDIX B 


AIRCRAFT FIRE CONTROL DEVELOPMENTS IN THE AAF 1X1 


February 28, 1945 


1. Scope of Memo. 

1.1. This memo has been written in response to a re¬ 
quest, from Dr. E. L. Bowles, for suggestions concerning 
the organization, program, and staff of an aerial gunnery 
research unit at Eglin Field. 

The moment, however, one attempts to analyze this 
individual situation he becomes inescapably involved in 
the broader problem of the way in which the AAF as a 
whole provides for the development of new devices and 
techniques of aerial warfare. 

The body of the present memo is therefore devoted to 
an analysis of the broad problem; and the recommenda¬ 
tions relating specifically to gunnery research at Eglin 
are to be found in 5. 


2. The Functional Relation between the Steps Involved in 
Plane-to-Plane Control Problems. 

2.1. On the Flow Chart [given in Figure 1] is schemati¬ 
cally indicated the main sequence of the steps involved in 
the genesis and solution of the equipment problems in 
plane-to-plane fire control. The diagram is intended to 
illustrate the fact that the military problems arise from 
combat needs, and pass through a series of steps which 
culminate in new equipment in combat use. 

2.2. Air Warfare Analysis [AWA] occupies a central 
place in the diagram; and should in fact exercise a cen¬ 
tral and correlating function in the actual process. 

The AWA group should develop: 

a. Knowledge of the quantitative theory of the tactical 
and strategic problems of air warfare. 

b. Knowledge of the analytical theory of aircraft fire 
control equipment (sights, sighting mechanisms, 
bombsights, rocket sights, turret controls, etc.). 

c. Knowledge of the theory (probability and statistics) 
of obtaining hits in aerial gunnery; and of the pro¬ 
cedures for estimating the effectiveness of such hits 
(vulnerability theory). 

d. Knowledge of the psychological-physiological as¬ 
pects of the man-machine combinations in aerial 
gunnery (tracking, training, etc.). 

e. Correlated knowledge of all ORS activities. In fact, 
AWA should serve as a central group which unifies 
the activities of the separate Operation Research 
Sections; and should carry out comparative analyses 
of their separate findings. 

f. In addition to this large body of scientific and tech¬ 
nical knowledge, the AWA group should also have 


m Except for minor changes in form, which do not affect 
the subject matter, this is a copy of the original memorandum. 


an understanding of the practical problems involved 
in the introduction of new equipment (logistics, 
spare part and maintenance service, training, etc.). 
The purpose of AWA should be to provide the Chief of 
Air Staff with expert scientific advice on the employment 
of air power; to maintain current information on the 



Figure 1. Flow chart for aerial gunnery fire control 
devices. 

This chart is drawn for the case of aerial gunnery develop- 
ments. A closely similar chart exists for other types of items 
required in aerial warfare (rockets, guided missiles, bombs, etc.); 
and the AWA group should occupy the same position in all 
such charts. 

There is no intention to imply that at present, when AWA 
does not exist, there are no useful liaison cross-connections be¬ 
tween the main functions in the outside circle; nor would all 
of these liaison cross-connections necessarily function through 
AWA if this group existed. 


tactics and techniques of aerial gunnery, aerial bombing, 
etc.; to investigate the mathematical principles which 
underlie the conduct of aerial warfare; and to reduce those 
principles to a form which will assist the staff of a combat 
air force in its planning of operations. This may be more 
specifically stated as follows: 

A. To assist the Chief of Air Staff by 

a. Developing a scientific and quantitative basis for 
the operation of combat aircraft. 


CONFIDENTIAL 


223 





















224 


APPENDIX B 


b. Determining the relative desirability of 

(1) different categories and types of combat air¬ 
craft, 

(2) different equipment for a given combat air¬ 
craft, 

(3) different methods of operating combat air¬ 
craft, by the use of so much of this scientific and 
quantitative basis as has been developed. 

c. Making recommendations consequent to those 
determinations concerning 

(1) military characteristics of future military air¬ 
craft, 

(2) changes in characteristics of present aircraft, 

(3) changes in training, 

(4) changes in the operational use of aircraft. 

B. To determine what tests and studies are required 
to accomplish A, and to recommend where these should 
be carried out. 

C. To study the basic problems of bombardment 
aviation and to analyze the results of tests and studies 
made elsewhere. 

D. To conduct such tests and studies required to ac¬ 
complish A as the Chief of Air Staff may direct. 

2.3. The diagram recognizes a distinction between Proof 
Testing and Assessment of Performance, although the dia¬ 
gram also emphasizes that these two are closely inter¬ 
locked. 

The primary function of Proof Testing is to determine 
whether the equipment is operable under conditions of 
Service use, it being assumed that the quality of per¬ 
formance is assured by the design. The primary function 
of Assessment is to measure performance by quantitative 
tests, and to interpret the practical effectiveness of this 
performance. 

2.4. It is particularly important that the two-way com¬ 
munication, via the AWA group, be actively main¬ 
tained between Training, Assessment, and Combat. 

3. The Required Administrative Organization. 

The Flow Chart discussed above takes no account of the 
command function; nor of the practical problem of cogni¬ 
zances. But if the administrative set-up is to serve use¬ 
fully, it must be such that the various interchanges and 
main currents of flow of the diagram can proceed promptly. 

Examination of the diagram will serve to indicate some 
of the weaknesses of the present situation. The chief 
comments appear to be: 

3.1. The central correlating AWA group does not exist 
within the present structure of the AAF. 

3.2. The Development function is largely fulfilled by 
commercial groups operating under contract to Wright 
Field; and their links with the other functions of the 
diagrams are weak. 

3.3. The Assessment function cannot be carried out prop¬ 
erly unless a Gunnery Research Unit (GRU) is estab¬ 
lished and suitably supported. A fine nucleus for such 
a unit now exists at Eglin. 

3.4. The research aspects of the Training function ap¬ 
pear to be well provided for at Laredo. 


3.5. The Proof Testing function appears to be well pro¬ 
vided for at Eglin. 

3.6. The problem of cognizances is a serious one, and 
does not appear to be suitably solved at present. Thus 
certain functions listed on the chart come under M and 
S, others under 0 C and R; and the inter-relations are 
not clear, nor do they always appear to be well defined. 

3.7. The Assessment function cannot be carried out and 
utilized effectively unless tactical and strategic consider¬ 
ations are allowed to enter. This means one of two 
things: either the Gunnery Research Unit at Eglin must 
itself be permitted to cut across the preconceived com¬ 
mand functions, or these broader relationships must be 
assumed by some other group, working at a higher 
echelon. 

The Chart assumes the second of these two solutions. 
That is, it assumes that the AWA group would cut 
across command functions, and would appear on the 
AAF organization chart at the same level as Manage¬ 
ment Control. A close tie-up between Air Warfare 
Analysis and the Gunnery Research Unit would then 
assure that the latter had access to necessary informa¬ 
tion and to decision levels. 

4. Recommendations. 

4.1. It is recommended that an Air Warfare Analysis 
Section, as briefly described above, be set up. This 
group should be located in Washington, and should re¬ 
port at the same level as Management Control. More 
detailed suggestions as to staff and program can be 
furnished if desired. 

4.2. It is recommended that the existing program of 
gunnery research at Eglin Field be expanded, and 
organized into a Gunnery Research Unit, which is dis¬ 
cussed in more detail in 5. 

5. Gunnery Research Unit. 

5.1. Function and Program. 

The primary function of this unit should be to carry 
out such ground and air tests of aerial gunnery equip¬ 
ment and devices as will result in reliable quantitative 
measures of performance; and to carry out such analyses 
of these performance data as will result in the assess¬ 
ment of this performance in terms of practical end re¬ 
sults such as hits on enemy aircraft, probable number of 
enemy planes destroyed, etc. 

It is recognized that, in order to carry out this func¬ 
tion, GRU must maintain a program of fundamental 
research and development, both in analytical methods 
and in the instrumental aspects of the testing. 

The Gunnery Research Unit, occupying as it does a 
place just in advance of production, should furnish the 
quantitative and objective evidence on the basis of 
which the Command can make correct decisions. 

5.2. Staff and Materiel. 

It is clear that any estimates of staff and materiel will 
be subject to frequent modification. The following out¬ 
line, although made on a unit basis, is subject to the 
likelihood of frequent changes. There are five units: 
military administration, planning and analysis section, 


CONFIDENTIAL 



APPENDIX B 


225 


flight test section, bench test section, and experimental 
squadron (pretest). 

a. Military Administration. 

The size of the staff required here is a military decision. 
The function of the military administration is to pro¬ 
vide housekeeping and administrative services for the 
civilian and officer scientific and technical personnel. 

b. Planning and Analysis Section. 

This group would be responsible for the planning, 
direction and analysis of all the projects of the 
Gunnery Research Unit. The personnel and qualifica¬ 
tions required are: 

1 Director 
1 Associate 
Director 


2 Fire Control 
Statisticians 


2 Aerial Gunnery 
Mathematicians 

2 Instrument 
Engineers 
1 Aerodynamical 
Engineer 

1 Radar Engineer 
1 Psychologist- 
Physiologist 
4 Project 
Technical 
Representatives 

4 Computers 
Secretaries, 

Stenographers, 

Clerks, etc. 

c. Flight Test Section. 

This includes director, pilots, aircraft, cameras, gun¬ 
ners, engineers, computers, photographers, and shop 
facilities. 

d. Bench Test Section. 

This would include: 

1 “Texas Engine” with maintenance crew and shop 
facilities. 

2 Experimenters (at least one trained in Physics, at 
least one trained in Psychology, at least one trained 
in Statistics). 


Experience in fire control of 
Armament work, understanding 
of statistical principles, ability 
to plan and direct broad pro¬ 
grams of research. 

At least one with experience in 
hit-probability work, at least 
one with experience in experi¬ 
mental design. 

At least one with experience on 
lead computing sights, at least 
one with experience in ballistics. 
One with mechanical, the other 
with electrical, background. 
Attack curve theory — aerody¬ 
namical limitations on maneu¬ 
verability, etc. 


Understanding of the scientific 
techniques used in gunnery re¬ 
search, ability to direct projects 
to obtain the right data. 


3 Clerks 
4-6 Gunners. 

e. Experimental Squadron {Pretest). 

This unit would be concerned with preliminary tests 
of suggestions, gunnery equipment and gunnery 
tactics. It should include something like: 

2 heavy bombers (B-17 or -24 and B-29), 

2 medium bombers (B-25 or -26 and A-26), 

4-6 fighters (P-40, P-47, P-51 and P-61, possibly a 
P-80A and a “piggyback” P-38 or P-63). 

Air crews (The members of these air crews having 

(1) recent combat experience, 

(2) a willingness and liking for experiment, and 

(3) technical background and belief in gunnery re¬ 
search.) 

Ground crews. 

Intermediate shop facilities. 

Adequate photographic equipment. 

The members of these aircrews should be slowly ro¬ 
tated as adequately qualified personnel become avail¬ 
able after combat tours of duty. 

5.3. Comments. 

a. The GRU should be given the broad responsibility 
for developing and carrying out a basic program in 
aerial gunnery research. It must be given the author¬ 
ity to plan and develop this program; and although 
the unit would of course be expected to carry out 
needed tests, it must be protected from the necessity 
of accepting every directive that any one can think up. 

This is an essential point. What is needed here is 
an establishment carrying on a real scientific program, 
not a mail-order (or telephone) service station. The 
Unit should be assigned broad problems and fields of 
activities; but should not be required to accept, from 
non-scientific sources, directives on specific jobs. 

b. The organization and program must be maintained 
sufficiently flexible to meet new conditions as they 
inevitably arise. 

c. The Unit should be an independent Branch of the 
Proving Ground Command, and should not be 
subsidiary to some other Branch. 

d. The Office personnel should remain with this Unit 
for extended assignments; and should receive recog¬ 
nition for this type of service. 

e. Such compensation and status must be available 
for civilian scientists and engineers as will attract and 
hold men of high caliber. This definitely requires 
status and privileges equivalent to those enjoyed by 
officers. 

f. The Unit must have (preferably through such an 
organization as AW A) close liaison with the policy 
and decision levels of the AAF. 


CONFIDENTIAL 






BIBLIOGRAPHY 

Numbers such as AMP-502.1-M5 indicate that the document listed has been microfilmed and that its title appears 
in the microfilm index printed in a separate volume. For access to the index volume and to the microfilm, consult 
the Army or Navy agency listed on the reverse of the half-title page. 

Summary 

1. “Sequential Tests of Statistical Hypotheses,” Abraham Wald, The Annals of Mathematical 
No. 2, June 1945, pp. 117-186. 

2. Sequential Analysis of Statistical Data: Applications, H. A. Freeman and Abraham Wald, 

Columbia University Press, Revised Report 30.2R [OEMsr-618], SRG-C, Sept. 15, 1945. 


Statistics, Vol. XVI, 

prepared for AMP, 
AMP-21.1-M4 


PART I 
Chapters 1-8 

In references 53 through 116 the Study number is the AMP Study number and the Paper number is the contrac 
tor’s Paper number. 


AMP REPORTS, MEMORANDA, AND NOTES 

AMP Study 44: Training in Aerial Free Gunnery 

1. Mathematical Evaluation of Sighting Methods Used in 

American Free Gunnery Training, Churchill Eisenhart, 
OEMsr-1007, Memorandum 44.1M, AMG-C, Sep¬ 
tember 1943. AMP-502.1-M5 

AMP Study 55: Lead Computing Sights 

2. An Introduction to the Analytical Principles of Lead 

Computing Sights, Saunders MacLane, OSRD 4037, 
OEMsr-1007, Memorandum 55.1M, AMG-C, April 
1944. AMP-503.6-M21 

2a. Ibid., p. 11. 

AMP Study 57: Lead Computing Sights — Nose 
Attacks 

3. Errors Made by a Lead Computing Sight When the Target 
Follows a Pursuit Course, Walter Leighton, OSRD 3062, 
OEMsr-1007, Report 57.1R, AMG-C, December 1943. 

AMP-503.6-M8 

4. A Study of the Effect of Relative Wind and Gravity Upon 
a Caliber .50 Bullet in Relation to the Behavior of a Lead- 
Computing Sight, Walter Leighton, OSRD 3373, 
OEMsr-1007, Report 57.2R, AMG-C, March 1944. 

AMP-503.6-M15 

5. On Errors Made by a Lead-Computing Sight with Special 
Reference to Head-On Attacks, Walter Leighton, OEMsr- 
1007, Memorandum 57.1M, AMG-C, October 1943. 

AMP-503.6-M5 

6. A Note on Lateral and Vertical Deflections for a Caliber .50 

Projectile in Plane-to-Plane Fire, Walter Leighton, 
OEMsr-1007, Memorandum 57.2M, AMG-C, February 
1944. AMP-503.3-M2 

7. Addendum to AMP Report 57.1 R. Errors Made by a 
Lead-Computing Sight when the Target Follows a Pursuit 
Curve, Walter Leighton, OSRD 3374, OEMsr-1007, 
Memorandum 57.3M, AMG-C, March 1944. 

AMP-503.6-M13 

8. Pursuit Courses, Walter Leighton, OEMsr-1007, Memo¬ 
randum 57.4M, AMG-C, March 1944, p. 1. 

AM P-503.7-MI 

8a. Ibid., p. 10. 

8b. Ibid., p. 15. 

9. Solution of the Differential Equation: —a ( d\/dt) + (1 /u)r) 

= da/dt, Walter Leighton, OEMsr-1007, Memorandum 
57.5M, AMG-C, Mar. 29, 1944. AMP-503.6-M19 

10. Fighter Attacks at Low Rates of Closure, Revision of AMP 
Memorandum 57.6M, Walter Leighton and Charles 
Nichols, OEMsr-1379, Memorandum 57.7M, AMG-N, 
October 1945. AMP-504.4-M16 


AMP Study 85: Study of Fuze Dead-Time Correc¬ 
tions in AA Director 

11. The Prediction of Fuze Setting, F. J. Murray and Arthur 

Sard, OSRD 4057, OEMsr-1007, Report 85.1R, AMG-C, 
August 1944. AMP-704-M9 

12. The Smoothing Effect of a Follow-Up Motor, Arthur Sard, 

OEMsr-1007, Memorandum 85.1M, AMG-C, October 
1945. AMP-704-M12 

AMP Study 103: AAF Training Program 

13. Some Uses of Variable Speed Mechanisms in Fire Control, 

Magnus R. Hestenes, OEMsr-1007, Memorandum 
103.1M, AMG-C, Apr. 14, 1944. AMP-503.1-M3 

AMP Study 104: Analytical Assessment of Certain 
Lead Computing Sights 

14. An Analytic Study of the Performance of Airborne Gun- 

sights, Donald P. Ling, OSRD 5318, OEMsr-1007, 
Report 104.1R, AMG-C, June 1945. AMP-502.1-M25 

14a. Ibid., p. 46. 

15. Deflection Formulas for Airborne Fire Control, Magnus R. 

Hestenes, OSRD 6270, OEMsr-1007, Report 104.2R, 
AMG-C, October 1945. AMP-503.3-M8 


15a. 

Ibid., 

Chap. II. 

15b. 

Ibid., 

Chap. III. 

15c. 

Ibid., 

Chap. IV. 

15d. 

Ibid., 

Chap. V. 

15e. 

Ibid., 

p. 16. 

15f. 

Ibid., 

p. 134. 

15g. 

Ibid., 

pp. 141-143. 


16. The Theory of an Electromagnetically Controlled Hooke’s 
Joint Gyroscope, Donald P. Ling, OSRD 6228, OEMsr- 
1007, Report 104.3R, AMG-C, October 1945. 

AMP-502.1-M34 

17. Aerial Gunnery and Gyro Sights, A Manual for Gunners, 

Donald P. Ling, OSRD 6284, OEMsr-1007, Report 
104.4R, AMG-C, October 1945. AMP-502.12-M22 

18. The Optical System of the Mark 18 ( K-15) Gyro Gunsight, 

with an Appendix of the Tracing of Rays Through a Thick 
Lens or System of Lenses, L. Charles Hutchinson, 
OSRD 6283, OEMsr-1007, Report 104.5R, AMG-C, 
October 1945. AMP-502.12-M23 

19. Simple Formulas to Fit the Values Tabulated in the Firing 
Tables FT 0.50 AC-M-1, George Piranian, OEMsr-1007, 
Memorandum 104.1M, AMG-C, Apr. 6, 1944, p. 2. 

AMP-503.1-M2 

20. Ballistic Formulas, Alex E. S. Green, Memorandum 

104.2M, Research Division, U. S. Army, Army Air 
Forces Instructors’ School, published by AMP, NDRC, 
June 1944, p. 9. AMP-503.1-M6 

20a. Ibid., p. 5. 

21. Bias Errors of the K-Sand K-12 Sights, Irving Kaplansky 


CONFIDENTIAL 


227 


228 


BIBLIOGRAPHY 


and Mae Reiner, OEMsr-1007, Memorandum 104.3M, 
AMG-C, May 1945. AMP-502.11-M13 

21a. Ibid., p. d of Summary. 

21b. Ibid., Appendix V. 

22. The Combination of a Random and a Systematic Error, 

Arthur Sard, OEMsr-1007, Memorandum 104.4M, 
AMG-C, September 1945. AMP-502.141-M9 

23. A Proposed Vector-Rate Sight for Airborne Fire Control, 
Magnus R. Hestenes, OSRD 6274, OEMsr-1007, 
Memorandum 104.5M, AMG-C, October 1945. 

AMP-502.1-M33 

AMP Study 106: Aerodynamic Pursuit Curve 

24. Aerodynamic Pursuit Curves for Overhead Attacks, 

George H. Handelman, and W. Prager, OSRD 4019, 
OEMsr-1066, Report 106.1R, AMG-B, Aug. 7, 1944, 
Appendix A. AMP-503.7-M6 

25. Aerodynamic Lead Pursuit Curves for Overhead Attacks, 
George H. Handelman, OSRD 6383, OEMsr-1066, Re¬ 
port 106.2R, AMG-B, Oct. 31, 1945. AMP-503.7-M15 

26. The Aerodynamic Pursuit Curve, M. M. Day and W. 

Prager, OEMsr-1066, Memorandum 106.1M, AMG-B, 
July 12, 1944. AMP-503.7-M4 

AMP Study 107: Determination of the Orientation of 
an Airplane 

27. The Stereographic Spherimeter, L. Charles Hutchinson 

and John H. Lewis, OEMsr-1007, Memorandum 107.1M, 
AMG-C, January 1945. AMP-503-M3 

28. Correction for Roll, Pitch and Yaw with the Spherimeter, 
L. Charles Hutchinson and John H. Lewis, OEMsr-1007, 
Memorandum 107.2M, AMG-C, February 1945. 

AMP-503-M4 

AMP Study 119: Use of Vector Sights in Plane-to- 
Plane Combat 

29. What Per Cent of Own Speed Deflection? Gustav A. Hed- 

lund, OSRD 4440, OEMsr-1007, Report 119.1R, 
AMG-C, November 1944, p. 101. AMP-503.3-M3 

29a. Ibid., Sec. 7. 

29b. Ibid., Sec. 10. 

30. Optimum Methods of Using Compensating Sights, Dan 

Zelinsky, OEMsr-1007, Report 119.2R, AMG-C, 
October 1945, pp. 32-36. AMP-502.12-M20 

30a. Ibid., p. 41. 

31. Average Percentages of Own Speed Deflection, Dan 
Zelinsky and M. J. Lewis, OEMsr-1007, Memorandum 
119.1M, AMG-C, January 1945, p. 14. AMP-503.3-M4 
31a. Ibid., pp. 16-19. 

31b. Ibid., pp. 8-10. 

31c. Ibid., p. 5. 

32. Position Firing Rules for the A-26, Dan Zelinsky, 

OEMsr-1007, Memorandum 119.2M, AMG-C, March 
1945. AMP-503.4-M7 

AMP Study 130: Collaboration with Radiation 
Laboratory on Aerial Gunnery Problems 

33. Ballistic Calibrations of Radar Range Aids to Airborne 

Cannon Fire, R. M. Thrall and George W. Mackey, 
OSRD 6288, OEMsr-1007, Report 130.1R, AMG-C, 
October 1945. AMP-503.1-M13 

AMP Study 142: Analytical Assistance to Project 14, 
Northwestern University (Section 7.2) 

34. Camera Evaluation of Bomber Gun Sights, A. A. Albert, 


OSRD 5441, OEMsr-1379, Report 142.1R, AMG-N, 
July 1945. AMP-502.14-M9 

AMP Study 143: General Electric Fire Control 

35. General Principles of the General Electric CFC Computer: 
Models 2CH1C1 and 2CH1D1, Magnus R. Hestenes, 
Daniel C. Lewis, and F. J. Murray, OSRD 5662, 
OEMsr-1007, Memo 143.1M, AMG-C, September 1945. 

AMP-503.5-M 14 

36. Gyroscopes of the General Electric CFC Computer in the 

B-29 Airplane, Magnus R. Hestenes, Daniel C. Lewis, 
and F. J. Murray, OSRD 5663, OEMsr-1007, Memo 
143.2M, AMG-C, September 1945. AMP-503.5-M12 

37. The Axis Converter and the Potentiometer Resolver in the 
General Electric B-29 Computer, Magnus R. Hestenes, 
Daniel C. Lewis, and F. J. Murray, OSRD 5664, 
OEMsr-1007, Memo 143.3M, AMG-C, September 1945. 

AMP-503.5-M13 

AMP Study 145: Optimum Gun Dispersion for the 
Mark 23 

38. Optimum Dispersion with the Mark 23 Fighter Gunsight, 

Wallace Givens, OEMsr-1379, Report 145.1R, AMG-N, 
September 1945. AMP-502.13-M21 

AMP Study 153: Computation of Aerodynamic 
Pursuit Courses 

39. Equations for Aerodynamic Lead Pursuit Courses, Leon 
W. Cohen, OSRD 5423, OEMsr-1007, Report 153.1R, 
AMG-C, July 1945, Appendix II. AMP-503.7-M12 
39a. Ibid., Sec. 7. 

39b. Ibid., Appendix I. 

39c. Ibid., Appendix II. 

40. Aerodynamic Lead Pursuit Courses, Leon W. Cohen, 

OSRD 5382, OEMsr-1007, Report 153.2R, AMG-C, 
July 1945. AMP-503.7-M11 

AMP Study 155: Calibration of the Mark 23 

41. Time of Flight Setting of a Lead Computing Sight, Irving 

Kaplansky, OEMsr-1007, Memo 155.1M, AMG-C, 
March 1945, p. 1. AMP-503.6-M38 

41a. Ibid., Sec. 5. 

AMP Study 157: Topics in Aerial Gunnery 

42. Offset Guns, Fighter Attack Against Bombers, Walter 
Leighton and Charles Nichols, OSRD 6210, OEMsr- 
1379, Report 157.1R, AMG-N, October 1945. 

AMP-504.4-M 17 

43. Simple Rules for Support Fire with a Vector Sight, Level 

and Related Attacks, Charles Nichols, OEMsr-1379, 
Memo 157.1M, AMG-N, May 1945. AMP-503.4-M8 

44. On Apparent Speed Firing, Charles Nichols, OEMsr- 

1379, Memo 157.2M, AMG-N, September 1945 (the 
late date of this paper is accounted for by recent Service 
return to the idea). AM P-503.4-M11 

Minutes of the Meeting of Joint Army-Navy NDRC 
Airborne Fire Control Committee for June 13, 1945, 
Saunders MacLane, July 18, 1945. 

AMP Study 160: Assessment of Fighter Sights 

45. Camera Assessment of Fighter Plane Gunsights, H. L. 

Garabedian, OSRD 6206, OEMsr-1379, Report 160.1R, 
AMG-N, October 1945. AMP-502.14-M14 

AMP Study 166: Study of Data Accumulated in 
Sight Evaluation Tests 

46. Results of a Recomputation of Sight Evaluation Test Data, 


CONFIDENTIAL 




BIBLIOGRAPHY 


229 


Wallace Givens, OEMsr-1379, Memo 166.1M, AMG-N, 
September 1945. AMP-502.141-M10 

47. Airborne Tracking and Ranging Errors, Arthur Sard, 
OEMsr-1007, Memo 166.2M, AMG-C, October 1945. 

AMP-503.2-M27 

AMP Study 167: Frangible Bullets 

48. Frangible Bullets and Aerial Gunnery , Gustav A. Hed- 

lund, OEMsr-1007, Memorandum 167.1M, AMG-C, 
July 1945. AMP-504.52-M5 

AMP Study 186: Harmonization Studies for the B-29 

49. Harmonization Studies for the B-29 Airplane, September 

1944 to April 1945, Philip Kissam, OSRD 5242, OEMsr- 
1365, Service Project AC-92, Report 186.1R, AMG-P, 
April 1945. AMP-502.2-M13 

AMP Study 188: Operational Problems of B-29 Fire 
Control 

50. Optimum Dispersion for Nose Turrets of a B-29, Arthur 

Sard and R. L. Swain, OEMsr-1007, Memorandum 
188.1M, AMG-C, October 1945. AMP-504.21-M15 

AMP Study 191: Camera Assessment of Airplane 
Skid 

51. Measurement of Angle of Attack and Skid in Rocket Fire 
Problems, H. L. Garabedian, OSRD 6203, OEMsr-1379, 
Report 191.1R, AMG-N, October 1945. 

AMP-502.14-M13 


MISCELLANEOUS AMP REPORTS 

52. Notes on Parameters of Probability Distributions, OSRD 
4045, AMP Note 13, with the assistance of Columbia 
University and Princeton University, June 1944. 

AMP-11-M2 

53. Tracking and the Fire Control Problem, Hassler 

Whitney, OSRD 5680, OEMsr-1007, AMP Note 21, 
AMG-C, September 1945. AMP-703.4-M12 

54. A Manual for the Use of Gnomonic Charts, A. A. Albert, 
OEMsr-1379, AMP Note 23, AMG-N, October 1945. 

AMP-503.1-M 14 

55. Scatter Bombing of a Circular Target, H. H. Germond and 

Cecil Hastings, Jr., OSRD 4572, OEMsr-818 and 
OEMsr-1007, Report 10.2R, AMG-C, and BRG-C, 
May 1944. AMP-803.4-M2 


APPLIED MATHEMATICS PANEL 
CONTRACTORS 7 PAPERS 

Applied Mathematics Group — Columbia (AMG-C) 

56. Diary of C. Eisenhart, G. A. Hedlund and R. M. Thrall, 

entitled, The Ballistics of Aerial Gunnery, Visit to Aber¬ 
deen Proving Ground August 25 and 26, 1944, OEMsr- 
1007, Study 104, Paper 258 (Revised), AMG-C, Sept. 
11, 1944. AMP-502-M3 

57. Absolute Angles of Attack of Enemy Fighter Aircraft, 

Churchill Eisenhart, OEMsr-1007, Study 44, Paper 213, 
AMG-C, June 29, 1944. AMP-503.8-M2 

58. Experimental Data for Certain Nose Attacks on B-29’s, 

E. Hewitt, OEMsr-1007, Study 188, Paper 475, AMG-C, 
Aug. 10, 1945. AMP-504.4-M15 

59. Experimental Verification of Optimum Percentages of 
Own Speed Lead, Arthur Sard and Dan Zelinsky, 


OEMsr-1007, Study 119, Paper 439, AMG-C, June 11, 
1945. AMP-503.3-M6 

60. An Empirical Verification of Position Firing, Herbert 
Solomon and Churchill Eisenhart, OEMsr-1007, Study 
44, Paper 323, AMG-C, Dec. 18, 1944. AMP-503.4-M6 

61. Position Firing Deflections Corresponding to Arbitrary 
Angles Off, Herbert Solomon and Churchill Eisenhart, 
OEMsr-1007, Study 44, Paper 322, AMG-C, Dec. 6, 

1944. AMP-503.4-M5 

62. A Note on Recalculated Position Firing Rules for the A-26, 

Dan Zelinsky, OEMsr-1007, Study 119, Paper 446, 
AMG-C, June 26, 1945. AMP-503.4-M10 

63. Position Firing Rules for Various Altitudes and Speeds, 

Samuel Eilenberg, OEMsr-1007, Study 119, Paper 422, 
AMG-C, May 23, 1945. AMP-503.4-M9 

64. Compensating Sights, Dan Zelinsky, OEMsr-1007, Study 
119, Chapter IV, Paper 499, AMG-C, Sept. 20, 1945. 

AMP-502.1-M32 

65. Bill Airs His Views, Fighter Attack Against Bombers, A 
Description of the Company Front Attack, Saunders 
MacLane, OEMsr-1007, Paper 362, AMG-C, Jan. 30, 

1945. AMP-504.4-M12 

66. Mechanization of Own Speed Sights when Used on a Nose 
Gun for Support Fire Against Frontal Parallel Attacks, 
Gustav A. Hedlund, Study 119, Paper 248, AMG-C, 
Aug. 25, 1944. 

67. The Sperry K-Sights, L. Charles Hutchinson, OEMsr- 
1007, Study 104, Paper 206, AMG-C, June 22, 1944. 

AMP-502.11-M3 

68. Calibration of Time of Flight, I Bomber Sight (superseded 
by Memorandum 155.1M), Irving Kaplansky, Study 
104, Paper 301, AMG-C, Oct. 30, 1944. 

AMP-503.6-M38 

69. Slewing Systems for Gyro Sights, Saunders MacLane, 

OEMsr-1007, Study 46, Paper 45, AMG-C, Sept. 6, 
1943. AMP-503.6-M4 

70. Amplification of Tracking Noise in the Gunsight Mark 18, 

Donald P. Ling, OEMsr-1007, Study 104, Paper 372, 
AMG-C, Feb. 21, 1945. AMP-503.2-M23 

71. Sperry Sights K-3, K-4 , and K-12, L. Charles Hutchin¬ 
son, OEMsr-1007, Study 104, Paper 236, AMG-C, July 
26, 1944 (contains a list of pertinent Sperry documents). 

AMP-502.11-M5 

72. Aerial Gunnery and Gyro Sights: A Manual for Gunners, 
Donald P. Ling, OSRD 6284, OEMsr-1007, Study 
104, Paper 489, AMG-C, October 1945. 

AMP-502.12-M22 

73. The Rate Sight K-15 (Mark 18), A General Discussion, 

Donald P. Ling, OEMsr-1007, Study 104, Paper 500, 
AMG-C, Jan. 31, 1945. AMP-502.12-M21 

74. The Optical System of the K-3 and K-4 Sights, L. Charles 

Hutchinson, OEMsr-1007, Study 104, AMG-C, July 
18, 1944. AMP-502.11-M4 

75. The Tracing of Rays Through a Thick Lens or Lens 
System, Application to the Mark 18, (superseded by 
AMP Report 104.5R), L. Charles Hutchinson, OEMsr- 
1007, Study 104, Paper 348, AMG-C, Jan. 10, 1945. 

AMP-900-M9 

76. A Method in Connection with the Mark 18 Sight, Donald 

P. Ling, OEMsr-1007, Study 104, Paper 209, AMG-C, 
June 27, 1944. AMP-503.2-M11 


CONFIDENTIAL 



230 


BIBLIOGRAPHY 


77. Solution of the Equations for the Behavior of the Mark 18 

Gunsight when Tracking an Arbitrary Space Course, 
Donald P. Ling, OEMsr-1007, Study 104, Paper 238, 
AMG-C, July 28, 1944. AMP-503.2-M11 

78. Deflection Formulas for Gun Sights of the Mark 18 Type , 

Donald P. Ling, OEMsr-1007, Study 104, Paper 358 
(Rev.), AMG-C, June 26, 1945. AMP-503.3-M7 

79. Calibrations for Straight Line and Pursuit Courses, 

Irving Kaplansky, OEMsr-1007, Study 155, Paper 390, 
AMG-C, Mar. 21, 1945. AMP-503.7-M10 

80. Trail and Gravity Offsets for a Modified Turret , Irving 

Kaplansky, OEMsr-1007, Study 155, Paper 365, 

AMG-C, Feb. 8, 1945. AMP-502.1-M21 

81. The Mark 18 in a Watermelon Turret , Irving Kaplansky, 

OEMsr-1007, Study 155, Paper 396, AMG-C, Apr. 3, 
1945. AMP-502.12-M17 

82. Trail Offsets of the Mark 18 in a Displaced Turret , Irving 

Kaplansky, OEMsr-1007, Study 155, Paper 383, 

AMG-C, Mar. 13, 1945. AMP-502.12-M15 

83. A Model Calibration for the Mark 21 and Mark 23, Irving 

Kaplansky, OEMsr-1007, Study 155, Paper 320, 

AMG-C, Nov. 28, 1944. AMP-502.13-M6 

84. The Irrelevance of Angle of Attack for the Mark 23, Irving 

Kaplansky, OEMsr-1007, Study 155, Paper 386, 

AMG-C, Mar. 16, 1945. AMP-502.13-M14 

85. Fire Control System of B-29 Tactics, Diary, GE Visit 

June 9 to 10, 194-4, Irving Kaplansky and Magnus R. 
Hestenes, OEMsr-1007, Study 104, Paper 197, AMG-C, 
June 26, 1944. AMP-502.2-M9 

86. A Proposal for Controlling the Speeds of the Total Cor¬ 

rection Motors in the General Electric 2CH1C1 Computer, 
Magnus R. Hestenes, Daniel C. Lewis, and F. J. Murray, 
OEMsr-1007, Study 188, Paper 453, AMG-C, July 5, 
1-945. AMP-503.5-M9 

87. Remarks with Regard to Modifications of the Present B-29 

Computer, Magnus R. Hestenes and Daniel C. Lewis, 
OEMsr-1007, Study 188, Working Paper 455, AMG-C, 
July 5, 1945. AMP-503.5-M10 

88. Facts Important in the Tactical Use of the B-29 Fire Con¬ 
trol System, Daniel C. Lewis and F. J. Murray, OEMsr- 
1007, Study 143, Paper 413, AMG-C, May 10, 1945. 

AMP-502.2-M15 

89. Summary of Results of Testing a B-29 Fire Control Com¬ 

puter Total Correction Motor, Daniel C. Lewis, OEMsr- 
1007, Study 143, Working Paper 409, AMG-C, May 4, 
1945. AMP-503.5-M6 

90. Constants for the Total Correction Motors in the 2CH1C1 

Computer for the B-29 Airplane, Daniel C. Lewis and 
F. J. Murray, OEMsr-1007, Study 143, Paper 494, 

AMG-C, Sept, 24, 1945. AMP-503.5-M15 

91. Ranging in Defense of the B-29 Against Nose Attacks, 

Daniel C. Lewis, OEMsr-1007, Study 188, Paper 394R, 
AMG-C, June 30, 1945. AMP-503.2-M25 

92. Sundial Analysis — Mathematical Comments on a Memo¬ 

randum of E. W. Paxson, Daniel C. Lewis and John H. 
Lewis, OEMsr-1007, Study 107, Paper 143, AMG-C, 

Mar. 31, 1944. AMP-13-M11 

93. Application of the Air Mass Coordinate Method to Aerial 
Gunnery Assessment, P. A. Smith, OEMsr-1007, Study 
187, Paper 471, AMG-C, July 27, 1945. 

AMP-502.1-M29 


94. Remarks on Skid in a Fighter Plane, Hassler Whitney, 

OEMsr-1007, Studies 124, 146, 153, and 164, Paper 
418, AMG-C, May 18, 1945. AMP-601.2-M13 

95. Roll, Pitch, Yaw Correction by Table, R. L. Swain, 

OEMsr-1007, Study 166, Paper 433, AMG-C, June 4, 
1945. AMP-502.142-MI 

96. Local Stabilization of Coordinates, P. A. Smith, OEMsr- 
1007, Study 166, Paper 417, AMG-C, May 16, 1945. 

AMP-503.1-M11 

97. Gyro Measurement of Rotations, P. A. Smith, OEMsr- 
1007, Study 166, Paper 405R, AMG-C, June 7, 1945. 

AMP-502.142-M2 

98. The Assessment of Gun-Camera Trials, Rollin F. Bennett 
and Arthur Sard, OEMsr-1007, Study 166, Paper 370, 
AMG-C, and Paper 440, SRG-C, Mar. 28, 1945. 

AMP-502.14-M6 

99. Remark on a Question in Probability, George W. Mackey, 

OEMsr-1007, Study 166, Paper 485, AMG-C, Aug. 29, 
1945. AMP-502.141-M8 

100. Preliminary Analysis of the S-4 Sight, E. R. Lorch and 

Dan Zelinsky, OEMsr-1007, Study 104, Paper 451, 
AMG-C, July 7, 1945. AMP-502.13-M 19 

101. Lead Formulas for the Fairchild S-3 Sight, Magnus R. 

Hestenes and Dan Zelinsky, OEMsr-1007, Study 104, 
Paper 363, AMG-C, Feb. 7, 1945. AMP-502.13-M9 

102. Gun Roll in the S-3, Dan Zelinsky, Study 104, Paper 376, 

AMG-C, Mar. 1, 1945. AMP-502.13-M 11 

103. More on the Gyro Error in the S-3, E. R. Lorch, OEMsr- 
1007, Study 104, Paper 419, AMG-C, May 21, 1945. 

AMP-502.13-M 16 

104. The Sperry S-8B Stabilized Sight, Samuel Eilenberg, 

OEMsr-1007, Study 104, Paper 420, AMG-C, May 22, 
1945. AMP-502.13-M 17 

105. Diary, Visit to Radiation Laboratory, Radar for Airborne 
Fire Control, Saunders MacLane, OEMsr-1007, Study 
104, Paper 155, AMG-C, May 29, 1944. 

AMP-502.1-M 11 

106. The AN/APG System No. 16, Visit to Sperry Gyroscope 
Company, Garden City, L. /., R. M. Thrall, OEMsr-1007, 
Study 130, Paper 336, AMG-C, Dec. 18, 1944. 

AMP-503.2-M21 

107. Lead Computing Sights with Variable Parameter “a”, 
Magnus R. Hestenes and Saunders MacLane, OEMsr- 
1007, Study 72, Paper 79, AMG-C, Oct. 28, 1943. 

AMP-503.6-M41 

108. How to Put the Target on the Spot. Are the Spots before 
Your Eyes Red or Weissf Saunders MacLane, OEMsr- 
1007, Study 55, Paper 190, AMG-C, June 2, 1944. 

AM P-503.6-M23 

109. Proposal for a Sight for Turrets with Velocity Tracking, 

Hassler Whitney, OEMsr-1007, Study 63, Paper 210, 

AMG-C, June 22, 1944. AMP-503.2-M10 

Applied Mathematics Group — Northwestern 
(AMG-N) 

110. Computations for Dofograph Ballistic Correction Charts, 

R. S. Wolfe, OEMsr-1379, Study 152, Paper 54, AMG-N, 
June 4, 1945. AMP-503.1-M 12 

111. Dispersion Patterns in Fire from Moving Aircraft, George 

Piranian, OEMsr-1379, Study 142, Paper 18, AMG-N, 
Dec. 5, 1944. AMP-502.141-M3 

112. On Apparent Speed Firing, Charles Nichols, OEMsr- 


CONFIDENTIAL 





BIBLIOGRAPHY 


231 


1379, Study 157, Paper 80, AMG-N, September 1945. 

AMP-503.4-M11 

Minutes of the Meeting of Joint Army-Navy NDRC Air¬ 
borne Fire Control Committee for June 13,1945, Saunders 
MacLane, AMG-N, July 18, 1945. 

113. Computational Procedures and Forms for Camera As¬ 
sessment of Fighter Plane Gunsights, R. S. Wolfe, OEMsr- 
1379, Study 160, Paper 84, AMG-N, Sept. 5, 1945. 

AMP-502.14-M12 

114. Determination of Directions in Space by Photographs of 
Two or Three Fixed Points, A. A. Albert, Study 157, 
Paper 56, AMG-N, June 1945. 

115. Kinematic Lead under Evasive Action and Its Determina¬ 

tion by Photography from the Bomber (Preliminary Note), 
R. S. Wolfe, OEMsr-1007, Study 172, Paper 48, AMG-N, 
May 10, 1945. AMP-503.6-M39 

116. Computation of Single Shot Probabilities in Camera Sight 

Assessment, A. A. Albert, OEMsr-1379, Study 142, 
Paper 23, AMG-N, Jan. 5, 1945. AMP-502.14-M5 

Applied Mathematics Group — Brown (AMG-B) 

117. “Calculating Angle of Attack vs. Load Factor Curves 
from the Usual Performance Data Given on Enemy 
Planes” (Letter to Churchill Eisenhart), George H. 
Handelman, OEMsr-1066, Paper 78, AMG-B, May 24, 

1944. AMP-503.8-M1 

Applied Mathematics Group — Princeton (AMG-P) 

118. Introduction to the Aerial Gunnery Problems of AC-92, 
John W. Tukey, OEMsr-1365, Memorandum 2, En¬ 
closure 8 to Bulletin 2 (AC-92), AMG-P, Aug. 16, 1944. 

AMP-501-M4 

119. The Current Status of the Simplest Attackability Problem, 
John W. Tukey, OEMsr-1365, Memorandum 12 (FCR 
1128), Sec. IV, AMG-P, June 7,1945. AMP-504.4-M14 

Statistical Research Group — Columbia (SRG-C) 

120. Gun Operation with Lead Computing Sights, H. Hotelling, 
Study 46, Paper 3, SRG-C, October 1942. 

121. Relative Target Motion and Effective Dispersion, Rollin 
F. Bennett, OEMsr-618, Paper 398, SRG-C, Jan. 8, 

1945. AMP-502.141-M5 

Experimental Group — New Mexico 

122. Tests Related to the Defense and Tactical Use of the B-29, 

R. E. Holzer and others, OEMsr-1390, Service Project 
AC-92, Interim Report W/TR3, University of New 
Mexico, Nov. 15, 1944. AMP-504.41-M3 

123. The Effect of Radar Domes, Eagle Vane and Faired Dome , 

on the Operational Speed of the Modified B-29, Speed 
Reductions Caused by Open Bomb Doors at Operational 
Altitude of33,000feet (Memorandum to Warren Weaver), 
E. J. Workman, OEMsr-1390, Report W/M, University 
of New Mexico, Jan. 3, 1945. AMP-504.6-M4 

124. APG-5 Test with K-3 Sight in B-17 No. 6696 (Memo¬ 

randum from Warren Weaver), E. J. Workman, OEMsr- 
1390, Service Project AC-92, Report W/M, University 
of New Mexico, Jan. 2, 1945. AMP-502.11-M11 

Experimental Group — Mt. Wilson Observatory 

125. Offset Guns in Fighter Airplanes, OEMsr-1381, Technical 
Report 8, Mount Wilson Observatory, Jan. 15, 1945. 

AMP-504.4-M10 

Applied Mathematics Panel Informal Working Papers 

126. Notes on the Assessment of a Bomber's Defensive Fire, 


Warren Weaver, Working Paper 1, AMP, October 1944. 

AMP-504.1-M15 

127. Experimental Determination of Fluctuating and Steady 

Errors in Plane-to-Plane Fire, Warren Weaver, Working 
Paper 2, AMP, Nov. 4, 1944. AMP-502.2-M10 

128. The Effect of Quasi-Steady Errors on Bullet Density, 

Warren Weaver, OEMsr-1007, Working Paper 3, AMP, 
Paper 312, AMG-C, Nov. 14, 1944. AMP-504.1-M17 

OTHER NDRC REPORTS 

129. The Interception of a High Speed Bomber, Clifford R. 
Simms, Report to NDRC Section 7.2, The Jam Handy 
Organization, Inc., Detroit, Mich., March 1944. 

AMP-504.4-M6 

130. The Bank of an Airplane and Load Factor under Condi¬ 

tions of General Flight, William M. Borgman, Report to 
NDRC Section 7.2, The Jam Handy Organization, Inc., 
Detroit, Mich., June 30, 1944. AMP-504.6-M2 

131. Vector Gunsights and Assessing Cameras, OEMsr-991, 
Division 7 Report to the Services 96, The Jam Handy 
Organization, Inc., Sept. 30, 1945. 

131a. Ibid., pp. 8-9. 

132. A New Type of Lead Computing Sight, Preliminary Re¬ 

port, Lucien LaCoste, War Research Laboratory, Uni¬ 
versity of Texas, Apr. 23, 1943. AMP-503.6-M1 

133. A Device for Computing a Correction in the Kinematic 

Lead Computation or Lead Computing Sights [Part 1], 
Lucien LaCoste, Report to NDRC Section 7.2, War 
Research Laboratory, University of Texas, Feb. 26, 
1945. AMP-503.6-M37 

134. The Two Component M Correction for Kinematic Lead and 
a Possible Way of Adding It to the Fairchild S-3 Sight, 
Lucien LaCoste, Report to NDRC Section 7.2, War 
Research Laboratory, University of Texas, Mar. 31, 
1945. 

135. Theoretical Analysis of the Performance of Gunsights of the 
Mark 18 Type, M. Golomb and R. O. Yavne, Report 
330-1706-198, Franklin Institute, November 1944. 

136. Tracking Errors in Systems Using Velocity-Tracking and 
Aided-Tracking Controls with Direct and Lead Computing 
Sights, OEMsr-330, NDRC Division 7, Section 7.2, 
Report to the Services 78, February 1944. 

AMP-503.2-M3 

137. The Air Mass Coordinate Method of Aerial Gunnery 

Assessment, E. G. Pickels, NDRC Section 7.2, Feb. 15, 
1945. AMP-503.3-M5 

138. Fighter Gunnery and Assessment of Fighter Gunnery. 
Minutes of Meeting of Committee, July 11,1945, Saunders 
MacLane, OEMsr-1007, Joint Army-Navy-NDRC Fire 
Control Committee, AMG-C, Aug. 6, 1945. 

AMP-501-M10 

139. Modification of the MK. 18 Gyro Gunsight (Memorandum 
to Hassler Whitney), John B. Russell, July 14, 1944. 

AMP-502.12-M4 

140. Symposium on Confidential Airborne Electronic Equip¬ 
ment, MIT Radiation Laboratory, June 15-17, 1944. 

AMP-900-M6 

141. Pursuit Curve Characteristics (Graphs), N. U. Mayall, 

OEMsr-101, Mount Wilson Observatory, Carnegie Insti¬ 
tution of Washington, Apr. 5, 1944. MP-503.7-M2 


CONFIDENTIAL 



232 


BIBLIOGRAPHY 


ARMY REPORTS 

Aberdeen Proving Ground, Md. 

142. Firing Sidewise from an Airplane. L Theoretical Con¬ 
siderations , H. P. Hitchcock, Project RX 114, Report 
116, Ballistic Research Laboratory, Aberdeen Proving 
Ground, Aberdeen, Md., Aug. 12, 1938. 

143. The Effect of Yaw upon Aircraft Gunfire Trajectories, 
T. E. Sterne, Report 345, Ballistic Research Laboratory, 
Aberdeen Proving Ground, Aberdeen, Md., May 1,1943. 
143a. Ibid., p. 4. 

144. Analytical Trajectories for Type 5 Projectiles , T. E. 
Sterne, Report 346, Ballistic Research Laboratory, 
Aberdeen Proving Ground, Aberdeen, Md., Apr. 7, 
1943, p. 1. 

145. On Direct Firing Tables for Aircraft Gunnery, with 
Particular Reference to Caliber .50 AP M2 Ammunition, 
T. E. Sterne, Report 396, Ballistic Research Laboratory, 
Aberdeen Proving Ground, Aberdeen, Md., Sept. 2, 
1943, p. 4. 

145a. Ibid., p. 6. 

145b. Ibid., p. 9. 

146. Ballistic Coefficients of Small Arms Bullets of Current 
Production, B. G. Karpov, Report 478, Ballistic Re¬ 
search Laboratory, Aberdeen Proving Ground, Aber¬ 
deen, Md., Aug. 1, 1944. 

147. German 20-mm, 13-mm, and 7.92-mm Aircraft Ammuni¬ 
tion, B. G. Karpov, Memorandum Report 248, Ballistic 
Research Laboratory, Aberdeen Proving Ground, Aber¬ 
deen, Md., Nov. 19, 1943, p. 3. 

148. Trajectories in Air Coordinates for Caliber .50 AP M2 
Projectile Fired from Aircraft, L. B. C. Cunningham and 
F. John, Memorandum Report 279, Ballistic Research 
Laboratory, Aberdeen Proving Ground, Aberdeen, Md., 
Mar. 8, 1944. 

149. Siacci Functions for Caliber .30 Frangible Ball T44, Aber¬ 
deen Proving Ground, Aberdeen, Md., September 1944, 

p. 1. 

150. A Photometeoronic Method for Bomb Ballistics and for 
Measurements of the Flight Performance of Aircraft, 
BRL Report No. 279, Aberdeen Proving Ground, Aber¬ 
deen, Md., Sept. 26, 1942. 

Central School for Flexible Gunnery, AAF 

151. Own Speed Sights, A. E. S. Green, Research Bulletin 101, 
AAF Central School for Flexible Gunnery, July 1, 1944. 

152. Emergency Sighting Rules for Gunners on B-29 Bombers, 
A. E. S. Green, Research Bulletin 107, AAF Central 
School for Flexible Gunnery, July 4, 1944. 

153. Judgment of Aspect Angles, E. W. Ray, Research 
Bulletin 121, AAF Central School for Flexible Gunnery, 
Sept. 30, 1944. 

154. Bullet Dispersion for B-17 and B-24 Aircraft, Research 
Bulletin 123, Research Division, AAF Central School for 
Flexible Gunnery, Oct. 1, 1944. 

155. Judgment of Attack and Support Situations in the Air, 
Research Bulletin 134, AAF Central School for Flexible 
Gunnery, May 24, 1945. 

156. A Brief Survey of Simpler Types of Airborne Gunsights, 
Saunders MacLane, Study 104, Paper 289, Revised, 
AMG-C, Nov. 24, 1944. Also revised and printed as 


Research Memorandum 29 by AAF Central School 
for Flexible Gunnery, June 19, 1945. AMP-502.1-M18 

157. Use of Compensating Sights Including the Problem of 
Support Fire, P. W. Ketchum, Research Bulletin 135, 
AAF Central School for Flexible Gunnery, June 25,1945. 

158. Bullet Dispersions in the B-29 Aircraft, Research Bulletin 
1943, Research Division, AAF Central School for Flexi¬ 
ble Gunnery, August 1945. 

159. Offset Guns on Fighter Aircraft, V. G. Grove, Research 
Bulletin 141, Research Division, AAF Central School 
for Flexible Gunnery, Laredo, Texas, Aug. 21, 1945. 

160. The Contribution of Fighter’s Curvature and Mush to a 
Support Gunner’s Deflection, N. Coburn, Research Memo¬ 
randum 49, AAF Central School for Flexible Gunnery, 
Aug. 31, 1945^ 

Operations Analysis Division, AAF 

161. Gun Climb, Harmonization, and Bullet Pattern, W. L. 
Ayres, Operations Analysis Section, Eighth Air Force, 
Nov. 12, 1944. 

162. Support Fire with the K-13 Sight on the Waist Guns, W. L. 
Ayres and J. W. Odle, Operations Analysis Section, 
Eighth Air Force, Dec. 27, 1944. 

163. Support Fire Against Jet Aircraft, J. W. Odle, Opera¬ 
tions Analysis Section, Eighth Air Force, Apr. 21, 1945. 

164. Fighter Attacks with Offset Guns, W. L. Ayres, J. S. 
Jillson, and J. W. Odle, Operations Analysis Section, 
Headquarters, Eighth Air Force, Dec. 28, 1944. 

165. Analysis of Enemy Fighter Activity as Related to Losses 
and Damage Suffered by Heavy Bombers on 39 Missions 
during July, August and September 1943, Operational 
Research Section, Eighth Bomber Command, Nov. 3, 
1943. 

166. A Manual of Flexible Gunnery for Aircraft, Operations 
Analysis Section, Ninth Bomber Command (USAAF), 
Cairo, Egypt, October 1943. 

167. Report on Design of Hayward Position Firing Sight, with 
Instructions for Installation and Harmonizing, Far 
Eastern Air Forces Operations Analysis Report 31, 
Dec. 31, 1944. 

168. Flexible Gunnery Accuracy and Optimum Dispersion 
Fifteenth Air Force, C. P. Wells, Operations Analysis 
Section, Fifteenth Air Force, Mar. 19, 1945. 

169. A Lightweight Radar Computer Combination for the B-29 
RCT System, A. E. S. Green, Operations Analysis Sec¬ 
tion, Twentieth Air Force Headquarters, Oct. 23, 1944. 

AAF Proving Ground Command, Eglin Field, Fla. 

170. Vertical Deflection Analysis: Flexible Gunnery, E. W. 
Paxson, AAF Proving Ground Command, Eglin Field, 
Fla., Feb. 4, 1944, p. 4. 

171. Graphical Correction Techniques for Azimuth and Eleva¬ 
tion, E. W. Paxson and John Lewis, Assessment Memo 
3 (S.T. 2-44-22), AAF Proving Ground Command, 
Eglin Field, Fla., May 29, 1944. 

172. Calculation of Deflections-Bomber in Accelerated Flight, 
E. W. Paxson, Assessment Memo 11, (S.T. 2-44-22), 
AAF Proving Ground Command, Eglin Field, Fla., July 
13, 1944. 

173; Attacks Made by a Fighter with Oblique Guns; Pursuit 


CONFIDENTIAL 



BIBLIOGRAPHY 


233 


Curves; Fixed Oblique Gunnery, E. W. Paxson, AAF 
Proving Ground Command, Eglin Field, Fla., Mar. 11, 
1944. 

Other Army Sources 

174. Reports from Captured Personnel and Material Branch , 
Military Intelligence Division, U. S. War Department, 
by Combined Personnel of U. S. and British Services for 
Use of Allied Forces, Mar. 31, 1945. 

175. GAF Ideas on the “Company Front ” Attack, AAF In¬ 
telligence Summary 45-2. 

176. Jap Fighter Tactics with Inclined Guns, Headquarters 
AAF Intelligence Summary 45-2, Jan. 30, 1945. 

177. Final Report on Test of Evaluation of Aerial Guns and 
Gun Sights, Serial 2-44-22, AAF Board Project F3270, 
Jan. 29, 1945. 

178. Combat Maneuvers and Fighting Control, J. J. Driscoll, 
Headquarters Second Air Division, Eighth Air Force, 
April 1945. 

179. Gunner's Information File: Flexible Gunnery, Air Force 
Manual 20, May 1944. 


NAVY REPORTS 

180. Mathematical Analysis of Ordinary and Deviated Pursuit 
Curves, Carmichael, Rulon, Gillman, Gode, Project 
RM-6, Special Devices Division, BuAer, Navy Depart¬ 
ment, Sept. 15, 1944, p. 105. 

181. Experimental Determination of the Path of a Fighter Plane 
in Attacking a Bomber, Phase B, E. W. Paxson, Contract 
N166S-2052, Report to Special Devices Division, 
BuAer, Navy Department, The Jam Handy Organiza¬ 
tion, Inc., Detroit, Mich., May 22, 1944, Sec. 7. 

182. Notes for Refresher Trainees on Defensive Combat Ma¬ 
neuvers, E. V. Hardway, Free Gunnery Standardization 
Committee, Naval Air Operational Training Command, 
Jacksonville, Fla., Nov. 23, 1944. 

183. Position Firing, Free Gunnery Standardization Com¬ 
mittee, Navy Department, June 1944. 

184. Air Gunnery, The Free Gun, BuAer, Navy Department, 
May 1942. 

185. Gunsight Mark 21-23: Recommendation for Calibration of 
Control Circuits for Fighter Sight, D. D. Cody, 11-44- 
AAA, Lukas-Harold Laboratory, Re4d-25, Lukas- 
Harold Corporation, Nov. 18, 1944. 

186. Gunsight Mark 18 Final Recommendations for Composite 
K etc., D. D. Cody and F. T. Rogers, Jr., 8-44-T, Lukas 
Harold Laboratory, Re4d-25, Lukas-Harold Corpora¬ 
tion, Aug. 24, 1944. 

187. Calibration of Time of Flight for the Pursuit of a Target 
Flying a Circular Course, G. E. Albert, 12-44-D, Lukas 
Harold Laboratory, Re4d-25, Lukas-Harold Corpora¬ 
tion, Dec. 7, 1944. 

188. Experiments and Research on Methods of Attacking Large 
Bombers, CINCPAC-CINCPOA Translations, No. 2, 
Nov. 6, 1944. 

189. Reference Notes on Upward and Downward Inclined 
Fixed 20-mm MGs for Type 2 Land Recco Plane, 
CINCPAC-CINCPOA Translations, Nov. 6, 1944. 

190. Boresight Patterns for Fighter Airplanes with Discussion 
of Factors Affecting Aiming Allowances, Confidential 


Technical Note F-43, Reference Aer-E-3251-EMV, 
F41-3, VF, BuAer, Navy Department, June 4,1943, p. 3. 

191. Mark 18 Service Manual, BuOrd, Navy Department. 

BRITISH REPORTS 

192. Standard Corkscrew Maneuvers, Appendix A, OSRD 

WA-5195-9F, Letter BC/S.30343/Air/Ops. 1(c), Head¬ 
quarters Bomber Command, Great Britain, May 27, 
1945. AMP-504.1-M19 

193. Zone Shooting-Evasive Action, Report 71, RAF Air 
Fighter Development Unit, Apr. 30, 1943. 

194. Fixed Gun Air Firing, OSRD WA-4435-80, Report AP 
1730B, G 1, Department of the Air Member for Train¬ 
ing, Air Ministry, Great Britain, November 1943. 

AMP-503.4-M1 

195. General Notes on Gun Sighting, OSRD II-5-7214(S), Re¬ 

port A.P. 1730B, London, Air Ministry, Great Britain, 
September 1942, Vol. I, Chap. 1. AMP-502.1-M3 

196. A Note on Mark lie GGS Performance in Relation to Basic 

Ballistic Requirements, OSRD WA-4435-8K, Armament 
Department Note Arm. 226, Royal Aircraft Establish¬ 
ment, Great Britain, September 1943. AMP-502-M1 

197. A Method of Obtaining Operational Stability in AGL 
Mark I GGS Systems (Notes), OSRD WA-4435-8i, FC 
Memorandum 98, Armament Department, Royal Air¬ 
craft Department, Great Britain, July 1944. 

AMP-503.6-M25 

198. The Phenomenon of Aerial Aim Wander, An Essay on its 
Mathematical Description, Statistical Measurement, and 
Influence on Gunnery Performance, L. B. C. Cunning¬ 
ham, OSRD WA-2055-4a, AWA Report 51, Air Warfare 
Analysis Section, Great Britain, Apr. 29, 1944. 

AMP-502.1-M9 

199. The Service Trials on the Mk II Gyro Gunsight for 
Turrets, Melvill Jones, OSRD-II-5-5807(S), Final Re¬ 
port Ex/GRU/47, Great Britain, Apr. 8, 1943. 

AMP-502.12-MI 

200. The Stability of Blind Firing Systems, C. W. Gilbert, 
OSRD II-5-5927(S), Report GRU/M.8, Gunnery Re¬ 
search Unit, Exeter, Eng., Mar. 14, 1944. 

AMP-502.2-M4 

MISCELLANEOUS SOURCES 

201. Mathematics of the 3A2 Trainer, The Jam Handy 
Organization, Inc., Detroit, Mich., October 1944. 

202. Experimental Determination of the Path of a Fighter 
Plane in Attacking a Bomber: Phase B: Mathematical 
Analysis of Photographic Observations, E. W. Paxson, 
The Jam Handy Organization, Inc., USN Contract 
N 166S-2052, May 22, 1944. 

203. Elementary Mathematics of Aerial Gunnery, E. W. 

Paxson, The Jam Handy Organization, Inc., Detroit, 
Mich., June 1943. AMP-503.1-M1 

204. Tail Gun Computing Sight, Edmund B. Hammond, Jr., 
Sperry Gyroscope Co., Inc., Mar. 16, 1943. 

AMP-502.1-M4 

205. Theoretical Analysis of the Sperry-Draper Sight, K-Sight 
and Sperry Stabilized Sight, Sperry Gyroscope Co., Inc. 

AMP-502.1-M37 


CONFIDENTIAL 



234 


BIBLIOGRAPHY 


206. The K-3 and K-4 Aircraft Sight Error Analysis, Edmund 

B. Hammond, Jr., Document 5250-B-A, Sperry Gyro¬ 
scope Co., Inc., May 3, 1944. AMP-502.11-M2 

207. Steady State Prediction Error of the S-8 Sight , O. T. 

Schultz, Document 5252-2009, Sperry Gyroscope Co., 
Inc., Aug. 18, 1944. AMP-502.13-M3 

208. Handbook of Instructions, Operation, Service, and Over¬ 

haul: Central Station Fire Control, Computer Models 
2CH1C1 and 2CH1D1 for the B-29 Airplane , Reports 
GE-18787B and AN 11-70A-9, GEI-18787A, General 
Electric Co., June 10, 1945. AMP-503.5-M8 

209. General Electric Gyro-Stabilized Sight with Lead Control 

for Remote Turrets, J. R. Moore, Report TR-31298, The 
General Electric Co., Sept. 12, 1944. AMP-502.1-M14 

210. “Visierfragen” [Sight Questions or Points], Theodor 
Wilhelm Schmidt, Bericht der Lilienthal Gesellschaft, 
OSRD WA-5195-9C, Apr. 4, 1941, p. 151. 

AMP-502.1-M2 

211. “The Aerodynamics of a Spinning Shell/’ Fowler, 
Gallup, Lock, and Richmond, Philosophical Transac¬ 
tions of the Royal Society (London), Ser. A, 1920, Vol. 
221, pp. 295-387. 

212. “Ein Nomogramm fiir die Hohenabhangigkeit der Ge- 
schossflugdauer von Flugzeugwaffen/’Theodor Wilhelm 
Schmidt, Jahresbericht der Deutschen Luftfahrtforschung, 
1939, Vol. Ill, p. 157. 

213. Lehrbuch der Ballistik, C. Cranz, Julius Springer, Fifth 
Edition, Berlin, 1925, Vol. I, p. 137. Div. 3-220-M3 

214. Zur Ballistik des Querabschusses mit kleiner V 0 — Drall- 


oder leitwerkstabilisierte Geschosse bei Jdger-Viellauf- 
bewaffnung, Schtissler, Experiments and Memoranda 
2129, LFA Hermann Goring, Braunschweig, Nov. 1,1944. 

215. “Verfolgungskurven im Luftkampf/’A. Fricke, UM 650, 
Zentrale fiir Wissenschaftliches Berichtswesen der Luft¬ 
fahrtforschung des Generalluftzeugmeisters, Aug. 26, 1943. 

216. “Vorhaltetheorie fiir Bewegliche Flugzeugschausswaf- 
fen,” A. Fricke, UM 777, Zentrale fiir Wissenschaftliches 
Berichtswesen der Luftfahrtforschung des Generalluftzeug¬ 
meisters, June 29, 1944. 

217. Bericht S9, Lilienthal Gesellschaft, 1940. 

218. Bericht 153, Lilienthal Gesellschaft, Aug. 13-14, 1942. 
218a. Ibid., “Kreiselvisiere,” H. Kortum. 

219. [Windvane Sight, the own-speed sight], Anleitung fiir 
den Gebrauch des Windfahnenkornes, February 1916. 

AMP-502.1-M 1 

220. Fairchild Type S-3 Gyro Computing Sight, Fairchild 
Camera and Instrument Corp., Jamaica, N. Y., Sep¬ 
tember 1944. 

221. Interception and Escape Techniques at High Speed and 

High Altitudes, Model 416, W. B. Klemperer, Report 
SM-3263 (Revised), Douglas Aircraft Co., Inc., Oct. 
23, 1941, Appendix I. AMP-504.6-M1 

221a. Ibid., Appendix VI. 

221b. Ibid., Appendix VII. 

221c. Ibid., Appendix III. 

222. A General Survey of the Problems Entering into Plane-to- 

Plane Fire Control (Thesis), John M. Wuerth, Princeton 
University, May 1942. AMP-502.2-M1 


PART II 
Chapter 9 


1. The Sighting Problem for Airborne Fire Control, Hassler 

Whitney, OSRD 6276, OEMsr-1007, AMP Report 
124.1R, AMG-C, October 1945. AMP-601.2-M21 

la. Ibid., p. 25 

lb. Ibid., Secs. 8, 9 and 10. 

lc. Ibid., pp. 11, 12. 

l d. Ibid., pp. 14, 15. 

le. Ibid., Sec. 13. 

l f. Ibid., Secs. 13, 14. 

l g. Ibid., Appendix I. 

lh. Ibid., Sec. 32. 

li. Ibid., Secs. 16-36. 

2. Angular Rate Methods in Rocket Sighting, Irving Kaplan- 

sky, OSRD 6275, OEMsr-1007, AMP Report 124.2R, 
AMG-C, October 1945. AMP-601.2-M20 

2a. Ibid., Part IV. 

2b. Ibid., Part VI. 

2c. Ibid., Part V. 

3. Measurement of Angle of Attack and Skid in Rocket Fire 
Problems, H. L. Garabedian, OSRD 6203, OEMsr-1379, 
AMP Report 191.1R, AMG-N, October 1945. 

AMP-502.14-M13 

4. Airborne Fire Control, Summary Technical Report, 
NDRC Division 7, 1946, Vol. 3, Chap. 9. 

5. Trajectories of Aircraft Rockets 3.5” and 5.0”, OSRD 2225, 
OEMsr-418, CIT/UBC Report 27, CIT, Sept. 25, 1944. 

AMP-603-M1 

5a. Ibid., p. 76. 


6. The Testing of a Certain Range Finder, Hassler Whitney, 

OEMsr-1007, AMP Study 124, AMG-C Paper 305, 
Nov. 7, 1944. AMP-601.2-M2 

7. Notes on the Tracking Problem for Fighter Planes, Hassler 

Whitney, OEMsr-1007, AMP Studies 124 and 164, 
AMG-C Paper 329, Dec. 13, 1944. AMP-503.2-M20 

8. A Particular Method of Aiming Bombs and Rockets, 

Hassler Whitney, OEMsr-1007, AMP Studies 124 and 
164, AMG-C Paper 335, Dec. 15, 1944. AMP-601.2-M4 

9. The Mark 18 as a Range Finder, Harry Pollard, OEMsr- 

1007, AMP Study 124, AMG-C Paper 339, December 
1944. AMP-502.12-M10 

10. A Rocket Sight Called PARS, Hassler Whitney, OEMsr- 

1007, AMP Studies 124 and 164, AMG-C Paper 359, 
Jan. 27, 1945. AMP-601.2-M5 

11. On the Use of a Pendulum in Rocket Sighting from Air¬ 
craft, Hassler Whitney, OEMsr-1007, AMP Studies 124 
and 164, AMG-C Paper 381, Mar. 10, 1945. 

AMP-601.2-M7 

12. The Transient of a Single Gyro Sight with Fixed Sensi¬ 
tivity, Harry Pollard, OEMsr-1007, AMP Studies 124 
and 146, AMG-C Paper 406, Apr. 24, 1945. 

AMP-502.1-M22 

13. Formulas for Trajectory Drop and Flight Times for the 
5.0” HVAR and the 3.5” AR, Irving Kaplansky, OEMsr- 
1007, AMP Study 124, AMG-C Paper 414, May 10,1945. 

AMP-603-M3 


CONFIDENTIAL 



BIBLIOGRAPHY 


235 


14. Remarks on Skid in a Fighter Plane, Hassler Whitney, 

OEMsr-1007, AMP Studies 124, 146, 153, and 164, 
AMG-C Paper 418, May 18, 1945. AMP-601.2-M13 

15. A Suggestion for Calibrating PUSS, Irving Kaplansky, 

OEMsr-1007, AMP Study 124, AMG-C Paper 426, 
May 26, 1945. AMP-502.1-M24 

16. Relation between Skid and Forces Perpendicular to the 

Plane of Symmetry of an Aircraft, Dan Zelinsky, OEMsr- 
1007, AMP Studies 124 and 153, AMG-C Paper 430, 
June 1, 1945. AMP-504.6-M5 

17. The Trajectory Drop of Aircraft Rockets at Short Ranges, 

Hassler Whitney, OEMsr-1007, AMP Study 124, 
AMG-C Paper 431, June 2, 1945. AMP-603-M4 

18. Proposal of a Release Condition for Tossing Rockets, 

Donald P. Ling, OEMsr-1007, AMP Study 124, AMG-C 
Paper 434, June 5, 1945. AMP-606.1-MI 

19. A Remark on the Effect of Banking in the Rocket Sighting 

Problem, Harry Pollard, OEMsr-1007, AMP Study 124, 
AMG-C Paper 436, June 6, 1945. AMP-601.2-M18 

20. Tracking and the Fire Control Problem, Hassler Whitney, 

OSRD 5680, Study 63, AMP Note 21, AMG-C, Septem¬ 
ber 1945. AMP-503.2-M24 

21. A Theory of Toss Bombing, Harry Pollard, OSRD 6041, 

OEMsr-1007, AMP Report 146.1R, AMG-C, Septem¬ 
ber 1945. AMP-803.5-M12 

22. , The Use of H in Toss Bombing, Harry Pollard, OEMsr- 

1007, AMP Studies 146 and 164, AMG-C Paper 344, 
Dec. 29, 1944. AMP-803.5-M4 

23. Formulas Useful in Toss Rocketry, Donald P. Ling, 


OEMsr-1007, AMP Study 146, AMG-C Paper 401, 
Apr. 16, 1945. AMP-606.1-M2 

24. A Suggestion for Camera Measurement of Skid, Irving 

Kaplansky, OEMsr-1007, AMP Study 146, AMG-C 
Paper 473, July 31, 1945. AMP-502.14-M11 

25. A Modified Release Condition for Toss-Bombing, Harry 

Pollard, OEMsr-1007, AMP Study 146, AMG-C Paper 
476, Aug. 8, 1945. AMP-803.5-M11 

26. The Azimuth Problem in Toss-Bombing, Harry Pollard, 

OEMsr-1007, AMP Studies 146 and 164, AMG-C Paper 
495, Sept. 18, 1945. AMP-803.5-M13 

27. Toss-Bombing a Moving Target, Harry Pollard, OEMsr- 

1007, AMP Studies 146 and 164, AMG-C Paper 364, 
Feb. 7, 1945. AMP-803.5-M6 

28. Gravity Drop in a Two Component Fighter Sight, Irving 

Kaplansky, OEMsr-1007, AMP Study 164, AMG-C 
Paper 429, May 31, 1945. AMP-502.13-M18 

29. The Use of Range Rate in a Fighter Gun Sight, Irving 

Kaplansky, OEMsr-1007, AMP Study 164, AMG-C 
Paper 450, June 30, 1945. AMP-503.2-M26 

30. Notes on Pneumatic PUSS, [Part I], L. Charles Hutch¬ 

inson, OEMsr-1007, AMP Study 164, AMG-C Paper 
461, July 17, 1945. AMP-502.1-M28 

31. Notes on Pneumatic PUSS, [Part II], Rate of Climb 
Indicator, L. Charles Hutchinson, OEMsr-1007, AMP 
Study 164, AMG-C Paper 478, Aug. 10, 1945. 

AMP-502.1-M30 

32. An Introduction to the Analytical Principles of Lead 

Computing Sights, Saunders MacLane, OSRD 4037, 
OEMsr-1007, AMP Memorandum 55.1M, AMG-C, 
Mar. 31, 1944. AMP-503.6-M21 


PART III 
Chapter 10 


APPLIED MATHEMATICS PANEL REPORTS 
AND MEMORANDA ON STUDIES OF 
ANTIAIRCRAFT EQUIPMENT 

AMP Study 22: Test Firing 

1. Analysis of Trial Fire Methods, E. Bromberg, E. Paulson, 
and James J. Stoker, OSRD 4526, OEMsr-945, OEMsr- 
1007, and OEMsr-618, AMP Report 22.1R, AMG-C, 
AMG-NYU, and SRG-C, November 1944. 

AMP-702-M5 

2. Meteorological Test Firings (A Joint Project of the Army 

Ordnance Department and the Signal Corps), James J. 
Stoker, OEMsr-1007, AMP Memorandum 22.9M, 
AMG-C, September 1943. AMP-702-M4 

AMP Study 23: Muzzle Velocity 

3. Errors in Slant Range of the 90-mm Gun as Affected by 

Errors in Estimating Muzzle Velocity, E. Bromberg, 
E. Paulson, and James J. Stoker, OSRD 2053, OEMsr- 
1007 and OEMsr-618, AMP Report 23.1R, AMG-C and 
SRG-C, November 1943. AMP-703.6-M2 

AMP Study 25: AA Fire Control 

4. A Bibliography of AA Artillery, Daniel C. Lewis, 

OEMsr-1007, AMP Memorandum 25.2M, AMG-C, 
September 1944. AMP-701-M1 

5. Supplementary Bibliography of AA Artillery, Daniel C. 


Lewis, OEMsr-1007, AMP Memorandum 25.3M, 
AMG-C, June 1945. AMP-701-M2 

6. The Accuracy and Effectiveness of Antiaircraft Fire, 

Daniel C. Lewis, OEMsr-1007, AMP Memorandum 
25.4M, AMG-C, October 1945. AMP-700-M2 

AMP Study 29: Airplane Course Data 

7. The Mathematical Theory of the Hunt Miniature Range, 
E. J. Moulton and Daniel C. Lewis, OSRD 1604, 
OEMsr-1007, AMP Report 29.1R, AMG-C, July 1, 

1943. AMP-703.3-M2 
AMP Study 45: Stereo Movies of Tracers 

8. Tracer Stereographs, Daniel C. Lewis, OSRD 3212, 
OEMsr-1007, AMP Report 45.1R, AMG-C, January 

1944. AMP-703.5-M8 
AMP Study 85: Study of Fuse Dead-Time Correc¬ 
tions in AA Director 

9. The Prediction of Fuse Setting, F. J. Murray and Arthur 

Sard, OSRD 4057, OEMsr-1007, AMP Report 85.1R, 
AMG-C, August 1944. AMP-704-M9 

9a. Ibid., p. 17. 

10. The Smoothing Effect of a Follow-up Motor, Arthur Sard, 

OEMsr-1007, AMP Memorandum 85.1M, AMG-C, 
October 1945. AMP-704-M12 

AMP Study 99: Curved Flight 

11. Extrapolation by Least Squares with A pplication to A A 
Fire Control, Daniel C. Lewis, OEMsr-1007, AMP 


CONFIDENTIAL 



236 


BIBLIOGRAPHY 


Memorandum 99.1M, AMG-C, February 1945. 

AMP-703.6-M7 

12. On the Efficiency of the Curved Flight Director, Daniel 

C. Lewis, OEMsr-1007, AMP Memorandum 99.2M, 
AMG-C, September 1945. AMP-703.2-M16 

12a. Ibid., pp. 19-22. 

12b. Ibid., pp. 13-14. 

12c. Ibid., p. 35. 

12d. Ibid., pp. 41-43. 

12e. Ibid., pp. 30, 45. 

AMP Study 161: Analysis of Curved-Flight Data 

13. Analysis of Certain Data from the Bell Telephone Labora¬ 

tories on Curved Flight AA Directors, Kenneth J. Arnold, 
OEMsr-618, AMP Memorandum 151.1M, SRG-C, 
March 1945. AMP-703.2-M14 

Other AMP Papers 

14. An Exposition of Wiener's Theory of Prediction, Norman 

Levinson, OSRD 5328, OEMsr-1384, AMP Note 20, 
AMG-H, June 1945. AMP-13-M21 

15. The Combination of a Random and a Systematic Error, 

Arthur Sard, OEMsr-1007, AMP Memorandum 104.4M, 
AMG-C, September 1945. AMP-502.141-M9 

15a. Ibid., pp. 7-10. 

16. Results of a Recomputation of Sight Evaluation Test Data, 
Wallace Givens, OEMsr-1379, AMP Memorandum 
166.1M, AMG-N, September 1945, p. 16. 

AMP-502.141-M10 

17. Optimum Dispersion for Nose Turrets of a B-29, Arthur 
Sard and R. L. Swain, OEMsr-1007, AMP Memo¬ 
randum 188.1M, AMG-C, October 1945. 

AMP 504.21-M15 

18. Camera Evaluation of Bomber Gun Sights, A. A. Albert, 

OSRD 5441, OEMsr-1379, AMP Report 142.1R, 
AMG-N, July 1945. AMP-502.14-M9 

18a. Ibid., pp. 52-53. 

19. A Manual for the Use of Gnomonic Charts, A. A. Albert, 
OEMsr-1379, AMP Note 23, AMG-N, October 1945. 

AMP-503.1-M14 

20. Major H. F. Mitchell's Study of the M5 Director, Arthur 


Sard, OEMsr-1007, AMP Study 51, AMG-C, Oct. 8, 
1943. AMP-703.2-M6 

21. The Assessment of Gun-Camera Trials, Rollin F. Bennett 

and Arthur Sard (Appendix 1, “Vulnerability of Aircraft 
to Machine Gun Fire,” by Milton Friedman), OEMsr- 
1007 and OEMsr-618, AMP Report 166.1R, AMG-C 
and SRG-C, Mar. 28, 1945. AMP-502.14-M6 

21a. Ibid., pp. 17-22. 

22. Gyro Measurement of Rotations, P. A. Smith, OEMsr- 

1007, AMP Study 166, Revised, AMG-C Paper 405, 
June 7, 1945. AM P-502.142-M2 

23. Local Stabilization of Coordinates, P. A. Smith, OEMsr- 

1007, AMP Study 166, AMG-C Paper 417, May 16, 
1945. AMP-503.1-M11 

24. Roll, Pitch, Yaw Correction by Tables, R. L. Swain, 

OEMsr-1007, AMP Study 166, AMG-C Paper 433, 
June 4, 1945. AMP-502.142-MI 

25. Remark on a Question in Probability, George W. Mackey, 

OEMsr-1007, AMP Study 166, AMG-C Paper 485, 
Aug. 29, 1945. AMP-502.141-M8 

26. Possibility of Serial Correlation Increasing the Probability 

of at least One Hit, Rollin F. Bennett, OEMsr-618, Re¬ 
port 365, SRG-C, Nov. 25, 1944. AMP-502.141-M2 

27. Relative Target Motion and Effective Dispersion, Rollin 

F. Bennett, OEMsr-618, Report 398, SRG-C, Jan. 8, 
1945. AMP-502.141-M5 

28. Methods of Scoring Predictions by A A Directors, Kenneth 
J. Arnold and A. H. Bowker, OEMsr-618, AMP Study 
151, SRG-C Paper 527, July 5, 1945. AMP-703.2-M15 

29. A Warning About Statistical Efficiency, John W. Tukey, 
OEMsr-1365, Memorandum 17, AMG-P, Feb. 28, 1945. 

AMP-21-M3 

30. Methods for Computing Survival Probabilities, H. J. 
Greenberg and W. Kaplan, Report NR-1, July 1945. 
30a. Ibid., p. 5. 

31. The Probability of Survival of a Target on Any Course, 
When the Mean Points of Impact and the Joint Distribu¬ 
tion of the Errors Due to Tracking Are Known, H. B. 
Mann, Report NR-2, July 1945. 

32. Errors in Anti-Aircraft Fire-Control When the Target is 
Maneuvering, Report NR-3, November 1945. 


Chapter 11 


AMP REPORTS, MEMORANDA, AND NOTES 

1. Probability of Hitting a Twin Engine Airplane with 

Head-on AA Fire, E. Paulson, OSRD 1655, OEMsr-618, 
Navy Project NO-131, AMP Report 3.1R, SRG-C, 
February 1943. AMP-504.42-M4 

2. Increased Risk in Extending a Bombing Run (and Sup¬ 

plement to AMP Report 6.1R), W. A. Wallis, OSRD 1639, 
AMP Report 6.1R, and OSRD 1587, AMP Report 
6.2R, SRG-C, Jan. 26, 1943. AMP-705.1-M 1 

3. Probability that a IfAz' Rocket Fired from Astern will 
Destroy a Twin-Engine Bomber Ju-88, as a Function 
of Point of Burst, Milton Friedman, OSRD 4459, 
OEMsr-618, AMP Report 21.1R, SRG-C, July 1944. 

AMP-605-M2 

4. Optimum Burst Surface for JfAY Airborne Rocket Fired 
from Astern at Twin-Engine Bomber, Ju-88, Milton 


Friedman, OSRD 4961, OEMsr-618, AMP Report 21.2R, 
SRG-C, July 1944. AMP-605-M1 

5. Effectiveness of 4A" Airborne Rocket with T-5 Fuze when 

Fired at Twin-Engine Bomber from Astern, Milton Fried¬ 
man, OSRD 4960, OEMsr-618, AMP Report 21.3R, 
SRG-C, July 1944. AMP-605-M3 

6. Comparative Effectiveness of 5 " Shrapnel, Ifi-mm. H.E. and 

20-mm. H.E. Against Directly Approaching Aircraft, 
Milton Friedman, OSRD 1796, OEMsr-618, AMP Re¬ 
port 27.1R, SRG-C, August 1943. AMP-705.1-M3 

7. Comparative Effectiveness of 5" Shrapnel and 5" H.E. 

Against Directly Approaching Aircraft, Milton Friedman, 
OSRD 3098, OEMsr-618, AMP Report 27.2R, SRG-C, 
Dec. 28, 1943. AMP-705.1-M6 

8. Charts for Estimating Expected Number of Hits on a 
Directly Approaching Aircraft with 7.7-mm., 13.2-mm., 


CONFIDENTIAL 




BIBLIOGRAPHY 


237 


20-mm., and J+O-mm. AA Guns, Milton Friedman, and 
G. J. Stigler, OSRD 3532, OEMsr-618, AMP Report 
96.1R, SRG-C, April 1944. AMP-700-M1 

9. The Relative Effectiveness of Caliber .50, Caliber .60, and 
20-mm. Guns as Armament for Multiple Anti-Aircraft 
Machine Gun Turrets, Milton Friedman, OSRD 5388, 
OEMsr-618, AMP Report 140.1R (Revised), SRG-C 
and Ballistic Research Laboratory, January 1945. 

AMP-705.1-M7 

10. Estimates of the Vulnerability of the B-29 to Fighter Air¬ 
craft, Milton Friedman, OEMsr-618, AMP Memoran¬ 
dum 168.1M, SRG-C, February 1945. AMP-504.41-M5 

11. The Probability of Damage from Heavy Flak, M. M. Day, 

OSRD 5648, OEMsr-1066, AMP Report 185.1R, 
AMG-B, Sept. 1, 1945. AMP-504.42-M8 

12. Danger Area Around Standard Bombs, Milton Friedman, 

OEMsr-618, AMP Memorandum 194.1M, SRG-C, 
August 1945. AMP-504.41-M10 

13. The Probability of Damage to Aircraft Through Anti- 

Aircraft Fire. (The Dependence of Effectiveness on Shell 
Fragmentation Characteristics.), H. H. Germond; pre¬ 
pared for publication by B. H. Colvin, OSRD 5189, 
AMP Note 19, AMP, May 1945. AMP-504.42-M6 

BRITISH REPORTS 

14. Fragmentation and the Chances of Damage to Aircraft 
from AA Shells, E. S. Pearson, OSRD W-104-15, Report 


3, External Ballistic Department, Ordnance Board, 
Great Britain, April 1940. AMP-504.42-M1 

15. The Chances of Damage to Aircraft from AA Shells. A 

Generalization of Previous Methods of Solution, B. L. 
Welch, OSRD W-104-19, Report 23, External Ballistics 
Department, Ordnance Board, Great Britain, Aug. 8, 
1941. AMP-504.42-M3 

OTHER OSRD REPORTS 

16. The Probability of Damage to Aircraft Through Anti- 

Aircraft Fire, A Comparison of Fuzes when Used Against 
High Level Bombers Attacking a Concentrated Target, 
Garrett Birkhoff, Ward F. Davidson, D. R. Inglis, 
M. Morse, John von Neumann, and Warren Weaver, 
OSRD 738, July 16, 1942. AMP-704-M1 

17. Experiments on the Vulnerability of Military Aircraft to 
High-Explosive Shell Fragments, Report 205 of Section T, 
OSRD, September 1944. 

ARMY REPORTS 

18. Flak Analysis, Technical Manual 4-260, War Depart¬ 
ment, May 1, 1944. 

19. Formulas for the Effect of A A Fire Against an Enemy 
Airplane, M. Morse and W. R. Transue, Report 38, 
Technical Division Ballistic Section, Office of the Chief 
of Ordnance, July 1944. 


PART IV 
Chapter 12 


1. Theory of Games and Economic Behavior, John von 
Neumann and Oskar Morgenstern, Princeton Uni¬ 
versity Press, 1944. 

2. Legons Sur la Theorie Mathematique de la Lutte pour la 
Vie, V. Volterra, Gauthier-Villars, Paris, 1931. 

3. Elements of Physical Biology, A. J. Lotka, Williams and 
Wilkins, Baltimore, Md., 1925. 

4. Verifications Experimental de la Theorie Mathematique 
de la Lutte pour la Vie, G. F. Gause, Hermann et Cie, 
Paris, 1935. 

5. The Optimum Ammunition for the Rear Guns of a Bomber 

Usually Attacked by a Single-Engine Fighter, J. Wolfo- 
witz, OEMsr-1066, AMP Memorandum 2.6M, SRG-C, 
Nov. 6, 1942. AMP-504.4-M4 

6. Long-Range Weather Forecasting for Military Purposes 
(Parts I and II), AMP Report 1.1R, SRG-P, October 
1943. 

6a. Ibid., Part II, p. T-196. 

7. A Note on Certain Aspects of the Methodology of Opera¬ 
tional Research (Enclosure A to Naval Attache), P. M. S. 
Blackett, OSRD Ref. No. Loga B-7464A, London Re¬ 
port 2177, May 1943. 

8. “Aircraft in Warfare: The Dawn of the Fourth Arm, 
Paper V — The Principle of Concentration,” Frederick 
William Lanchester, Engineering, Vol. 98, Oct. 2, 1914, 
pp. 422-23. 


9. A Quantitative Aspect of Combat, B. O. Koopman, OSRD 
1874, OEMsr-1007, AMP Note 6, AMG-C, August 1943. 

AMP-900-M2 

10. The Mathematical Theory of Air Combat, L. B. C. 

Cunningham, OSRD II-5-1042, Great Britain [February 
1940]. AMP-504.1-M2 

11. An Analysis of the Performance of a Fixed-Gun Fighter, 
Armed with Guns of Different Calibres, in Single Home- 
Defense Combat with a Twin-Engine Bomber, L. B. C. 
Cunningham, E. O. Cornford, W. Rudoe, and J. Knox, 
OSRD WA-382-4d, AW A Paper 1, Air Warfare Analysis 
Section, Great Britain, February 1940. AMP-504.1-Ml 

12. Outline of Cunningham Papers, Milton Friedman, 

OSRD 1650, OEMsr-618, AMP Report 2.1R, 
SRG-C, Aug. 22, 1942. AMP-504.1-M3 

13. The Mathematical Theory of Air Combat, OSRD 1621, 
OEMsr-618, AMP Report 2.2R, SRG-C, Aug. 26, 1942. 

AMP-504.1-M4 

14. Comparative Effectiveness of .50" and 20-mm. Fighter 

Armament Against A Twin-Engine Bomber from Astern, 
OSRD 3701, OEMsr-618, AMP Report 2.5R, SRG-C, 
May 1944. AMP-504.4-M8 

15. On the Optimum Ammunition Mixture for a Fighter At¬ 

tacking a Multi-Engined Bomber, J. Wolfowitz, OEMsr- 
618, AMP Memorandum 2.11M, SRG-C, Dec. 26, 
1942. AMP-903.2-MI 


CONFIDENTIAL 



238 


BIBLIOGRAPHY 


16. Optimum Interrelation of Aiming and Dispersion Errors, 

J. Wolfowitz, OEMsr-618, AMP Memorandum 2.13M, 
SRG-C, Jan. 16, 1943. AMP-504.1-M6 

17. The Optimum Interrelation Between Gun and Aiming 
Errors when Several Shots are Fired From Each Position 
of Aim, AMP Memorandum 2.16M, SRG-C, June 1943. 

18. Evaluation of Conduct and Effectiveness of Air Operations 
in Pacific Ocean Area through 15 February 1945, Report 
6, AAF Evaluation Board, Pacific Ocean Area, April 
1945. 

19. [Gunnery and Bombing Tactics of B-29 Planes], Project 

AC-92, Bxdletins 1-13, compiled by Merrill M. Flood, 
OEMsr-1365, Service Project AC-92, FCR, AMG-P, 
July 1944 - February 1945. AMP-504.1-M14 

20. Tests Related to the Defense and Tactical Use of the B-29, 

R. E. Holzer, OEMsr-1390, Service Project AC-92, 
Report UNM/W/TR3, University of New Mexico, 
Nov. 15, 1944. AMP-504.41-M3 

21. Ground Tests of the B-29 Central Fire Control System, 

W. D. Crozier, OEMsr-1390, Service Project AC-92, 
Report UNM/W/TR5, University of New Mexico, 
Jan. 15, 1945. AMP-503.5-M3 

22. Experimental Observations on a Fighter's Ability to Main¬ 

tain a Consistent Aim During Attacks on High Speed 
Bombers, R. E. Holzer, OEMsr-1390, Service Project 
AC-92, Report UNM/W/TR6, University of New 
Mexico, Feb. 3, 1945. AMP-504.1-M18 

23. Experimentally-Determined Fighter Attack Courses 

Against a B-29, T. Zanstra, OEMsr-1380, Service Proj¬ 
ect AC-92, Report UNM/W-33, University of New 
Mexico, Apr. 24, 1945. AMP-504.41-M8 

24. An APG-5 Test with Mark IS ( K-15) Sight in B-17 Air¬ 

craft No. 42-4^39155, H. R. Snodgrass, OEMsr-1390, 
Service Project AC-92, Report UNM/W-34, University 
of New Mexico, Apr. 30, 1945. AMP-502.12-M18 

25. A Method of Analyzing Aerial Gun Camera Film Based 
on Use of Distant Reference Points, G. T. Pelsor, OEMsr- 
1390, Service Project AC-92, Report UNM/W-32, Uni¬ 
versity of New Mexico, Apr. 14, 1945. AMP-502.14-M7 

26. Studies of Defensive Fire Power of Formation of Airplanes, 
Final Report on Contract OEMsr-1381, Service Project 
AC-92, Mount Wilson Observatory of the Carnegie 
Institution of Washington, Aug. 31, 1945. 

AMP-504.21-M14 

27. General Description of the Method of Study of Defensive 
Fire Power of Formations of B-29 Airplanes Being Carried 
out at Mount Wilson Observatory, Pasadena, California, 
OEMsr-1381, Service Project AC-92, Technical Report 
3, Mount Wilson Laboratory, Oct. 17, 1944. 

AMP-504.21-M3 

28. The Methods of Study of a Squadron of Eleven Airplanes 

(Preliminary Report), OEMsr-1381, Technical Report 7, 
Mount Wilson Laboratory, Pasadena, Calif., Dec. 14, 
1944 - AMP-504.21-M6 


29. Analysis of Fire Power of a B-29 Airplane, OEMsr-1381, 
Service Project AC-92, Technical Report 1, Mount 
Wilson Laboratory, Pasadena, Calif., Sept. 16, 1944. 

AMP-504.21-M1 

30. Analysis of Fire Power of a Squadron of Four B-29 Air¬ 

planes, OEMsr-1381, Service Project AC-92, Technical 
Report 2, Mount Wilson Laboratory, Pasadena, Calif., 
Sept. 30, 1944. AMP-504.21-M2 

31. The Fire Power of a Squadron of Four B-29 Airplanes, 

Standard Mission II, OEMsr-1381, Service Project 
AC-92, Technical Report 4, Mount Wilson Laboratory, 
Pasadena, Calif., Nov. 6, 1944. AMP-504.21-M4 

32. The Fire Power of a Modified Squadron of Four B-29 Air¬ 

planes, OEMsr-1381, Service Project AC-92, Technical 
Report 6, Mount Wilson Laboratory, Pasadena, Calif., 
Nov. 24, 1944. AMP-504.21-M5 

33. Observational Results on the Defensive Fire Power of a 
Squadron of Eleven B-29 Airplanes, OEMsr-1381, Service 
Project AC-92, Technical Report 10, Mount Wilson 
Laboratory, Pasadena, Calif., Feb. 20, 1945. 

AMP-504.21-M7 

34. Observational Results on the Defensive Fire Power of a 

Squadron of Twelve B-29 Airplanes, OEMsr-1381, Tech¬ 
nical Report 11, Mount Wilson Laboratory, Pasadena, 
Calif., Feb. 20, 1945. AMP-504.21-M8 

35. Report of the Pasadena Conference on Defense of B-29 
Formations, Twentieth Air Force, AAF, Feb. 15-26,1945. 

36. Additional Measurements of the Defensive Fire Power of a 

Squadron of Twelve B-29 Airplanes, OEMsr-1381, Tech¬ 
nical Report 13, Mount Wilson Laboratory, Pasadena, 
Calif., Apr. 11, 1945. AMP-504.21-M9 

37. Preliminary Study of Fighter Attacks on a B-29, OEMsr- 
1381, Service Project AC-92, Technical Report 5, Mount 
Wilson Laboratory, Pasadena, Calif., Oct, 27, 1944. 

AMP-504.41-M2 

38. Duration of Strafing Attacks on a B-29 Aircraft, OEMsr- 
1381, Service Project AC-92, Technical Report 9, Mount 
Wilson Laboratory, Pasadena, Calif., Jan. 29, 1945. 

AMP-504.41-M4 

39. Evaluation of Fighter Attacks on B-29 Airplanes, OEMsr- 

1381, Technical Report 12, Mount Wilson Laboratory, 
Pasadena, Calif., Mar. 31, 1945. AMP-504.41-M7 

40. Offset Guns in Fighter Airplanes, OEMsr-1381, Technical 

Report 8, Mount Wilson Laboratory, Pasadena, Calif., 
Jan. 15, 1945. AMP-504.4-M10 

41. Foreword to Formulation of Manufacturing Specifications 
for Solid Propellants (Final Report), Raymond L. 
Arnett, OSRD 5851, Series P, No. 9, Alleghany Ballistics 
Laboratory, Division 3, Section H, NDRC. Nov. 1945. 

Div. 3-362-M3 

42. Battle Damage and Losses Connected with Enemy Fighter 
Activity, OAS Report 8, Bomber Command, Oct. 4, 1943. 


CONFIDENTIAL 





OSRD APPOINTEES 
APPLIED MATHEMATICS PANEL 

Chief 

Warren Weaver 

Deputy Chief 

Thornton C. Fry 

Acting Chief, May 29, 1945 to April 5, 1946 


Technical Aides 


B. H. Colvin 
H. H. Germond 
Cecil Hastings, Jr. 
Myrtle R. Kellington 
Margaret S. Piedem 


J. D. Williams 


D. C. Spencer 
S. S. Wilks 


Mina S. Rees 


I. S. SOKOLNIKOFF 


Members 


J. H. Dellinger 


L. J. Briggs 
R. Courant 


R. F. Mehl 
H. M. Morse 
P. M. Morse 


G. C. Evans 
L. M. Graves 


H. P. Robertson 
A. H. Tatjb 


0. Veblen 


CONFIDENTIAL 


239 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS FOR THE 

APPLIED MATHEMATICS PANEL 


Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-444 

The Franklin Institute 
Philadelphia, Pa. 

Technical Representative, 

H. B. Allen 

Computations. 

OEMsr-618 

Columbia University 

New York, N. Y. 

Official Investigator, 

H. Hotelling 

Director: 

W. Allen Wallis 

Statistical methods applied to air combat analysis, torpedo 
tactics, acceptance inspection, research and development, 
and related problems. 

OEMsr-817 

University of California 
Berkeley, California 
Technical Representative, 

J. Neyman 

Statistical analysis applied to bombing research concerned 
with problems of land mine clearance, the theory of pattern 
bombing and the bombing of maneuvering ships, and the 
theory of bomb damage. 

OEMsr-818 

Columbia University 

New York, N. Y. 

Technical Representative, 

J. Schilt 

Mathematical and statistical studies of bombing problems; 
the application of IBM computing techniques to statistical 
problems in warfare analysis. 

OEMsr-860 

Princeton University 

Princeton, N. J. 

Technical Representative, 

S. S. Wilks 

Statistical methods applied to miscellaneous problems in war¬ 
fare analysis and to (1) verification of various long-range 
weather forecasting systems; (2) a study of fire effect tables 
and diagrams for warships; (3) bombing accuracy studies, 
analysis of guided missiles, and the performance of certain 
heat-homing devices; and (4) the clearance of mine fields 
by explosive devices. 

OEMsr-944 

New York University 

New York, N. Y. 

Technical Representative, 

R. Courant 

Investigations in shock wave theory. 

OEMsr-945 

New York University 

New York, N. Y. 

Technical Representative, 

R. Courant 

Research in problems of the dynamics of compressible gases, 
hydrodynamics, thermodynamics, acoustics, and related 
problems. 

OEMsr-1007 

Columbia University 

New York, N. Y. 

Technical Representatives, 

E. J. Moulton 

S. MacLane 

A. Sard 

Miscellaneous studies in mathematics applied to warfare 
analysis with emphasis upon aerial gunnery, studies of fire 
control equipment, and rocketry and toss bombing. 

OEMsr-1066 

Brown University 

Providence, R. I. 

Technical Representative, 

R. G. D. Richardson 

Problems in classical dynamics and the mechanics of de¬ 
formable media. 

OEMsr-1111 

Institute for Advanced Study 
Princeton, N. J. 

Technical Representative, 

John von Neumann 

Studies of the potentialities of general-purpose computing 
equipment, and research in shock wave theory, with em¬ 
phasis upon the use of machine computation. 

OEMsr-1365 

Princeton University 

Princeton, N. J. 

Technical Representative, 

Merrill M. Flood 

Coordination of activities under Project AC-92 at the Uni¬ 
versity of New Mexico, Carnegie Institution of Washington 
at Pasadena, Columbia University, and Brown University. 

OEMsr-1379 

Northwestern University 
Evanston, Ill. 

Technical Representatives, 

E. J. Moulton 

Walter Leighton 

Studies in aerial gunnery, particularly the camera assessment 
of the performance of sights and of airplanes. 


240 


CONFIDENTIAL 












CONTRACT NUMBERS, CONTRACTORS AND SUBJECTS OF CONTRACTS (< Continued ) 


Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-1381 

Carnegie Institution of Washington Studies and experimental investigations in connection with 

Pasadena, Calif. the defensive fire power of various bomber formations by 

Technical Representative, means of model planes with their guns replaced by suitable 

Walter S. Adams light sources, the total fire power being estimates of the 

light intensity. 

OEMsr-1384 

Harvard University 

Cambridge, Mass. 

Technical Representative, 

Garrett Birkhoff 

Studies of the principles which determine the dynamic be¬ 
havior of a projectile entering water and the application of 
these principles quantitatively to the prediction of under¬ 
water trajectories and ricochet. 

OEMsr-1390 

The University of New Mexico 
Albuquerque, N. M. 

Technical Representative, 

E. J. Workman 

Studies and experimental investigations in collaboration with 
the Army Air Forces of the most effective formations and 
flight procedures for the B-29 airplane. Emphasis, originally 
upon the tactical use of the B-29, was later changed to a 
study of the defense of the B-29. 

Transfer of 
Funds 

National Bureau of Standards 

Computations by the Mathematical Tables Project for various 
agencies concerned with war research. 


CONFIDENTIAL 


241 






SERVICE PROJECT NUMBERS 


The projects listed below were transmitted to the Office of the Executive Secretary, OSRD, from the War or Navy 
Department through either the War Department Liaison Officer for NDRC or the Office of Research and Inventions 
(formerly the Coordinator of Research and Development), Navy Department. 


Service 

Project Number 

Subject 

AC-27 

AC-91 

AC-92 

ARMY PROJECTS 

Design data for bombardier’s calculator. 

Statistical problems of combat bombing accuracy. 

Collaboration of the NDRC with the AAF in determining the most effective tactical application of the 
B-29 airplane (continuing under AAF Proving Ground Command, Fire Power Analysis Project). 

AC-95 

AC-109 

AC-115 

AC-122 

AN-23 

CE-33 

OD-143 

OD-179 

OD-181 

QMC-35 

QMC-38 

QMC-43 

SC-81 

SC-100 

SOS-2 

Analysis of Waller trainer film. 

Textbook on flexible gunnery. 

Study of data accumulated in sight evaluation tests. 

Study of gun camera film scoring in order to devise a scoring computer. 

Studies of HE-IB attack on precision target. 

Checking of hydraulic tables. 

Study of fuze dead-time correction in AA director. 

Statistical assistance in rocket propellant tests and specifications. 

Study of relative destructive effect of machine gun fire against airplane structures. 

Food storage data statistics. 

Studies of various statistical problems encountered at the Climatic Research Laboratory. 

Statistical consultation for Quartermaster Corps inspection service. 

Rapid solution of linear equations with up to twenty-six unknowns. 

Binomial distribution calculations. 

Probability theory of balloon barrages. 


N-110 

N-112 

NAVY PROJECTS 

Mathematical studies of lead-computing sights for use with gunnery training. 

Study and evaluation of sighting methods of instruction used in U. S. Naval Aviation free gunnery 

N-120 

NA-167 

NA-177 

NA-195 

training. 

Preparation of instruction course for quality control and statistically based sampling procedures. 

Study of nozzle design for jet motors. 

An analytical method of determining ships’ speeds in turns from photographs of ships’ wakes. 

Study of jet propulsion devices operating at subsonic and supersonic velocities [continuing under 
Contract NOa(s)-7370]. 

ND-2 

NO-108 

NO-130 

NO-131 

Assistance to the Air Technical Division — studies of aircraft weapon effectiveness. 

Probability and statistical study of plane-to-plane fire. 

Air testing of Mark 15 bombsight. 

Probability studies desired in connection with estimating hits made by close-range AA gun fire at 
head-on airplane targets. 

NO-136 
NO-145 
NO-145 Ext. 
NO-145 Ext. 
NO-158 
NO-161 

Mathematical studies of dive-bomber and bomb trajectories in connection with Alkan dive-bombsight. 
Mathematical studies of bombing. 

Train probability calculations for bombing, November 1944. 

Probability curves for use in connection with gunnery salvo fire, June 1945. 

Antitorpedo-harbor defense nets. 

Theoretical studies of water entry phenomena [continuing under Contract NOa(s)-7370 with New 

York University and under Navy Contract with Harvard University} 

NO-188 

NO-206 

NO-237 

Study of torpedo spreads and their use against maneuvering targets. 

Studies of acceptance tests on ordnance material. 

Determination of depth of underwater explosions from surface observations [continuing under Con¬ 

NO-261 

tract NOa(s)-7370]. 

Statistical analysis of the data on thermal characteristics of targets and the relative performance of 
candidate heat-homing equipment. 


242 


CONFIDENTIAL 






SERVICE PROJECT NUMBERS ( Continued ) 

Service 

Project Number 

Subject 

NO-264 

NO-269 

NO-270 

Gun equilibrators. 

B-scan radar plotting device. 

Computation services (continuing under a transfer of funds from the Office of Research and Inven¬ 
tions to the Bureau of Standards). 

NO-272 

NO-280 

NO-294 

NR-101 

NR-105 

NS-165 

NS-166 

NS-302 

NS-364 

Computation of dynamic performance of A A computer (continuing under Contract NOrd-9153). 
Statistical assistance in rocket propellant tests and specifications. 

Study of tactical utilization of offset guns in fighter aircraft. 

Probability study of a proposed type of antiaircraft projectile. 

Fire effect tables (continuing under Contract NOrd-9240). 

Nonlinear mechanics. 

Gas globe phenomena in underwater explosions. 

High-temperature metals. 

Investigation of wave patterns created by surface vessels (continuing under Contract NOa(s)-7370). 


CONFIDENTIAL 


243 









INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 

For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


ABC method of eye shooting, 59 
Acceleration correction for target path, 
25-26 

Aerial gunnery research unit, recom¬ 
mendations for formation, 223- 
225 

Aerial warfare, general theory, 197-215 
factors affecting warfare problems, 
203-207 

“military worth,” 200-203 
offensive and defensive warfare, 197- 
199 

quantitative analysis, World War II, 
207-215 

Aeroballistics, 9-21 

aeroballistics versus classic ballistics, 
9 

ballistic deflections, 13-15 
dispersion, 18-20 
motion, small of projectile, 15-17 
time of flight, 9-13 
Air density, 38 

Airborne fire control, assessment 
see Fire control, analytic aspects 
Airborne rockets, 125-142 
advantages of rockets over shells, 125 
as risk to a bomber, 186 
fin-stabilized rockets, 126-127 
program summarized, 125-126 
Airborne rockets, aerodynamic aspects, 
127-130 

angle of attack, 127-129 
datum line, 128 
dive angle, 128 
flight angle, 128-129 
indicated airspeed, 128 
skid, 127-128 

tracking with a fixed sight, 129-130 
zero lift line, 128 

Airborne rockets, aiming problem, 130- 
132 

kinematic lead, 131-132 
lead in vertical plane, 130-131 
range to impact point, 132 
Airborne rockets, computing sights, 
140-142 

see also Sighting mechanism, airborne 
rockets 

peanut (automatic release sight), 142 
sight neglecting kinematic lead, 141- 
142 

sight predicting full lead in vertical 
plane, 140-141 
Airborne rockets, lead 
see Lead, airborne rockets 


Aircraft risk from flak, 167-194 
see also Antiaircraft risk computa¬ 
tions 

antiaircraft fire, physical process, 
168-170 

basic problem, mathematical analy¬ 
sis, 167 

fragmentation and damage calcula¬ 
tions, 189-194 

historical summary of studies, 181- 
189 

mathematical analogue, 170-181 
validity of studies, 194 
Aircraft risk from single shot, 181-189 
airborne rocket fire study, 186-187 
British studies, 182-183 
projectiles compared, 187-189 
risk to large group of aircraft, 185- 
186 

time and proximity fuzes compared, 
183-185 

Airspeeds, 38-39 
Algebraic sight, K-ll, 54 
Amplification factor in tracking errors, 
62 

AN/APG radar sets, 113 
Angle of attack, effect on pursuit 
curves, 37-39 
airspeed, 38-39 
deviation function, 37 
errors of approximation, 39 
load, 38 

qualitative effect, 39 
trajectory shift, 37 

Angular gravity drop of fin-stabilized 
rockets, 127 

Angular momentum of target, 26 
Angular parallax correction, 83-84 
Angular rate deflection, lead computing 
sights, 57-58 

Antiaircraft equipment, 145-166 
dry run errors, 150-158 
fire control systems, 145-150, 162- 
164 

guns, 145 

kinematic models for training pur¬ 
poses, 166 

linear and quadratic directors, 158- 
162 

numerical differentiation and smooth¬ 
ing, 166 

prediction circuits, 164-165 
tracer stereographs, 166 
trial fire methods, 165-166 
turret installations, 188 


Antiaircraft fire, physical process, 168- 
170 

fragments, 169-170 
fuze action, 169 

mechanism of antiaircraft guns, 168- 
169 

projectile launching, 168-169 
shell burst, 169-170 

Antiaircraft fire control systems, 145- 
150, 162-164 
assessments, 149-150 
components, 145-146 
determination of errors in system, 
146 

gun dispersion errors, 163-164 
input and output, 148 
muzzle velocity, 163 
overall-effectiveness, 146-147 
position, 147 

predicted position errors, 162 
predictor, 147-149 
probability calculation, 150 
tracking, 148 
tracking errors, 162 

Antiaircraft risk computations, 170- 
183 

British risk studies, 182-183 
complexity of problem, 170 
conditional probability of damage, 
176-179 

coordinate systems, 170-172 
distribution of shell bursts, 172-176 
probability of damage to single air¬ 
craft, 179-181 

steps in computing damage proba¬ 
bility, 168 

“Apparent speed” method of eye- 
shooting, 59 

Approach angle prediction, 45 

Armament of heavy bombers 

see B-29, defensive armament studies 

Assessment programs, airborne fire 
control 

see Fire control, analytic aspects 

Astrometric methods of space path de¬ 
termination, 96 

Automatic weapons, antiaircraft, 145, 
188, 189 

Axis converter in 2CH computer, 88 

B-29, defensive armament studies, 209- 
215 

facilities, 212 

historical background, 209-210 
optical studies at Pasadena, 213-215 

245 


CONFIDENTIAL 


246 


INDEX 


organizational background, 210-212 
problem, 212 
results, 213 
techniques, 212-213 
vulnerability of different parts, 188, 
189 

Backing-up process, dry run errors cal¬ 
culation, 156 

Ball-cage integrator for K-3 sight, 66- 
67 

Ballistic coefficient C, 12 
Ballistic deflections, 13-15 

angle subtended by gravity drop, 13 
dofographs, 15 

interpretation and calculation, 13-15 
lead computing sights, 57 
trial angle, 13 

Ballistic unit of 2CH computer, 87 
Ballistics, classical exterior versus aero- 
ballistics, 9 

Bearing determination methods, 96 
Bifurcated pursuit, 32-35 
Bombers, armament 

see B-29, defensive armament studies 
British antiaircraft risk studies, 182- 
183 

effect of shell burst density on proba¬ 
bility of damage, 182-183 
effectiveness of different antiaircraft 
shells, 182-183 

British automatic gun laying turrets 
(AGLT) Mark I, 114 
Burst of bullets theory, 18 
Burst surface of antiaircraft shells, 173- 
176 

Calibration 

gyroscopic sights in fighters, 79 
lead computing sights, 64-65 
2CH computer, 92 

Cameras as airborne assessment in¬ 
struments, 96-97 
Central station fire control, 82-93 
angular ballistic and parallax cor¬ 
rections, 83-84 
basic elements, 82 
follow-up system, 84-85 
kinematic deflection, 84 
remote control, 82 
time of flight, 84 
trajectory equations, 83 
2CH computer, 85-92 
Circular Gaussian distribution of 
bursts, 18 

Class A errors, fire control, 94-95 
Class B errors, fire control 
computing sights, 65-66 
definition, 55 

Fairchild S-4 sight system, 109-110 
gyroscopic lead computing sights, 76- 
78 


mechanical errors of lead computing 
sights, 68-69 
own-speed sight, 55 
steady errors, 95 

Collision course fighter attack, 120 
Computing sights for airborne rockets, 
140-142 

peanut (automatic release sight), 142 
sight neglecting kinematic lead, 141- 
142 

sight predicting full lead, vertical 
plane, 140-141 

Contact fuzes, risk computations, 176 
Continental ballistic coefficient C, 12 
Coordinate systems 

antiaircraft risk computations, 170- 
172 

azimuth and elevation system, 28 
gun elevation and traverse system, 
28 

plane of action system, 28 
relative coordinates for pursuit 
curves, 31 

rotation of coordinates, 97-99 
Siacci coordinates, 10-12 
sight elevation and traverse system, 
28 

stabilization of coordinates, 98 
Corkscrew combat maneuver, 43 

Damage from antiaircraft bursts, prob¬ 
ability study, 176-179 
see also Flak analysis 
angular density of fragments, 177- 

178 

angular fragmentation pattern, 178- 

179 

area of density of fragments, 177 
expected number of damaging hits, 
179 

fall-off law of fragment effectiveness, 
178-179 

fragmentation pattern of shell, 177- 
179 

probability of damage to target, 179 
shell and target characteristics, 176- 
177 

target speed, 177 
zone count of fragments, 178 
Damping of oscillatory tracking, 61 
Damping yaw, 15-16 
Data reduction, fire control 

see Fire control, raw data reduction 
Datum line of airborne rockets, 127- 
128 

Dead time error in antiaircraft fire con¬ 
trol systems, 163 
Defensive air warfare, 197-198 
Deflection, accelerated target, 25-26 
acceleration correction, 25-26 


kinematic and ballistic decomposi¬ 
tion, 26 

Deflection, nonaccelerated target, 23-25 
basic formula, plane of action, 23 
kinematic and ballistic decomposi¬ 
tion, 24-25 
leads, 23-24 

time of flight multiplier, 25 
tracking rate formulation, 24 
Deflection against pursuit curves 
see Own-speed sights, deflection 
against pursuit curves 
Deflection calculation, 99 
Deflection theory, 22-29 
ballistic deflections, 13-15 
conditions for validity, 22 
deflection defined, 22 
gun-roll error, 28-29 
necessity for systematic deflection 
theory, 22 
perfect bullet, 22 
special coordinate systems, 28 
timeback method of deflection calcu¬ 
lation, 29 

vector methods of general formula 
derivation, 27-28 

Density of bursts in antiaircraft fire, 
172-173 

Destruction probability, targets, 152 
Deviated pursuit 
defined, 30 

deflection percentage factor, 46 
Deviation function, pursuit curves, 37 
Dispersion, 18-20 

air-firing versus ground-firing pat¬ 
terns, 18-20 

errors in fire control, 94-95 
fin-stabilized rockets, 127 
forward fire distortion, 20 
harmonization, 20 
Dispersion pattern, 18-20 
muzzle velocity effect, 20 
optimum size, 103-106 
statistical description, 18-20 
theory, 18 

Disturbed reticle principle in lead com¬ 
puting sights, 58 

Dive angle of airborne rockets, 128 
Dofographs (ballistic-deflection charts), 
15 

Drag on bullet, 10 
Draper-Davis sight, 140-141 
Drift (bullet motion), 17 
Dry run errors, 150-158 
comparison index, 152 
definition, 150 

dynamic tester findings, 155-156 
expected value estimates, 152 
planning of dry runs, 150 
probability of shot with given dry 
run destroying target, 153-155 


CONFIDENTIAL 



INDEX 


247 


probability theory, 151 
serial correlation, 152-153 
survival probability, 152, 155 
unconditional vulnerability, 151 
Dry run errors, calculation, 156-158 
backing-up process, 156 
error perpendicular to relative tra¬ 
jectory, 156-157 

stabilization, observations made from 
rotating gun platform, 157-158 
Dry run principles in airborne assess¬ 
ment tests, 95 

Dynamic tester, dry run errors, 155- 
156 

Eddy current sights 
see Gyroscopic lead computing sights 
Effective launcher line for fin-stabilized 
rockets, 126 

Electromagnetic system, gyros, 74-75 
Elephant method of eye shooting, 58- 
59 

Error functions for linear and quadratic 
directors, 159-161 

Exponential smoothing circuit, 60-64 
aided tracking, 64 
amplification of tracking errors, 62 
ball-cage integrator as circuit, 66-67 
damping oscillatory tracking, 61 
decay of false leads, 61-62 
lead computation delay, 63-64 
operational stability, 62-63 
sight parameter choice, 63 
slewing routines, 61-62 
weighted averages, 61 
Exponential spot sights, 115 
Eye shooting estimation methods, 58- 
59 

ABC method, 59 
“apparent speed” method, 59 
disadvantages, 59 
early use, 58 
elephant method, 58-59 
EZ40-EZ45 German gyro sights, 80 

Fairchild S-4 sight system, 108-110 
circuit components, 108-109 
class B errors, 109-110 
complete circuit equations, 109-110 
Fall-off law for fragment effectiveness, 
178-179 

Fighter-plane armaments, quantitative 
analysis, 208-209 
Fin-stabilized rockets, 126-127 

angular gravity drop and time of 
flight, 127 
dispersion, 127 
effective launcher line, 126 
launchers, 126 
projectile speeds, 127 


Fire control, analytic aspects, 94-106 
airborne tests, 94 
current assessment methods, 95 
error classifications, 94-95 
instrumentation, 96-97 
measures of effectiveness, 101-103 
optimum dispersion pattern, 103-106 
summary, 106 

target path determination, 96-97 
Fire control, instrumentation, 96-97 
air mass coordinate technique, 97 
astrometric methods, 96 
bearing and range determination, 96 
distant reference point method, 97 
synchronization, 96-97 
Fire control, raw data reduction, 97- 
103 

deflection calculations, 99 
measure of effectiveness, 101-103 
parallax correction, 99 
probability affected by measurement 
errors, 99-101 

rotation of coordinates, 97-99 
Fire control errors 

classification by cause, 94-95 
statistical classification, 95 
Fire control systems, antiaircraft 
see Antiaircraft fire control systems 
Fire control theory, 107-121 

aided tracking on line of sight, 116— 
117 

exponential spot sights, 115 
new tactics, 119-121 
own-speed plus rate mechanization, 
117-118 

radar gunnery aids, 112-115 
research recommendations, 107 
sight parameter as a function of time, 
115 

stabilized sight systems, 107-112 
target course curvature correction 
mechanisms, 117 
tracking equation, 118-119 
Flak analysis, 189-194 

see also Damage from antiaircraft 
bursts, probability study 
flak charts, 190-192 
flakometers, 192-193 
group probability factors, 193-194 
methods of reducing total risk, 189- 
190 

problems to be studied, 189 
Flight angle of airborne rockets, 128- 
129 

Follow-up system for central station 
fire control, 84-85 

Fragmentation, antiaircraft fire, 169- 
170 

Fragmentation, flak analysis 

see Damage from antiaircraft bursts, 
probability study; Flak analysis 


Fragmentation patterns, antiaircraft 
shells, 177-179 

Frangible Ball T-44 projectile, 12 
Future range, definition, 10 
Fuzes 

antiaircraft shells, 169-170 
contact fuzes, risk computations, 176 
proximity fuzes, performance errors, 
163 

time and proximity fuzes compared, 
183-185 

Gaussian distribution, shell bursts, 18, 
173 

German gyro sights, 80 
Glook (mechajiism for mimicking air¬ 
craft orientation), 98 
Gnomonic projection computer, 98 
Gravity drop angle, 13 
Gun dispersion, 163 
Gun-roll errors, 28-29 
Gyroscope, stabilization use, 107 
Gyroscope unit of 2CH computer, 89- 
90 

Gyroscopic lead computing sights, 69- 
80 

class B errors, 76-78 
fighter sights, 78-80 
kinematic deflection production, 70- 
71 

optical system, 72-74 
physics of a gyro system, 71-72 
single gyroscopic eddy current sights, 
69-70 

turret types, 78 

Gyroscopic lead computing sights, me¬ 
chanics, 74-76 
electrical circuit, 75-76 
electromagnetic system, 74-75 
Hooke’s joint, 74 
torque, 74 

Gyroscopic lead computing sights in 
fighters, 78-80 

angle of attack, automatic computa¬ 
tion, 79-80 
calibration, 79 
circuit simplifications, 78-79 
German gyro sights, 80 
target course curvature, 79 

Handlebar motion, 64 
Hooke’s joint, gyro system, 74 
Hypothetical machine, air warfare 
analysis, 201-203 

Indicated airspeed (IAS) 
airborne rockets, 128 
formula, 38-39 
Initial speed of bullet, 9 
Initial yaw, definition, 9 


CONFIDENTIAL 



INDEX 




Instantaneous projectile speed of fin- 
stabilized rockets, 127 
Isogees (curves of equal load), 36 

Johns Hopkins University, target de¬ 
struction probability studies, 
154 

K-3 mechanical sight, 66-68 
ball-cage integrator, 66-67 
complete circuit, 67-68 
exponential smoothing, 66-67 
kinematic equations, 68 
lateral circuit, 67 

K-8 electrical lead computing sight, 80 
K-ll algebraic sights, 54 
K-12 mechanical sight, 68 
K-13 vector sight, 52-54 
Kinematic deflection 
fire control computer, 84 
formula, 24 

lead computing sights, 57 
production methods, 70-71 
Kinematic lead in airborne rockets, 
131-132 

Kinematic models for training pur¬ 
poses, 166 

Lag, 23-24 

Launchers for rockets, 126 
Lead, airborne rockets, 131-139 
dependence on weather conditions, 
133 

graphs of lead, 133-134 
kinematic lead computation^ 134-137 
kinematic lead in azimuth plane, 132 
kinematic lead in vertical plane, 131 
lead components, comparative im¬ 
portance, 133 

operation instability factor, 138 
sighting procedures for lead computa¬ 
tion, 137 

variables on which lead functions de¬ 
pend, 139 

Lead computing sights, 57-81 
apparent motion eye shooting, 58-59 
electrical types, 80 
gyroscope types, 69-80 
mechanical types, 66-69 
Lead computing sights, basic theory, 
57-66 

angular rate deflection, 57-58 
calibration concept, time of flight, 
64-65 

class B errors, 65-66 
conflict between ranging and tracking 
rates, 59-60 

decay of false leads, 61-62 
disturbed reticle principle, 58 
first-order nature, 57 


sight parameter, 60-61 
smoothing circuit, 60-64 
Lead equations, 23-24 
Lift coefficient of a bullet, 38 
Linear and quadratic directors, 158-162 
error prediction, 158-160 
random errors, 160-161 
theoretical performance, 161-162 
types of directors, 158 
Local stabilization, target coordinates, 
157 

Mach numbers, effect of high numbers 
on bomber attack, 43-44 
“ Macro-theory ” of warfare, 208 
Mathematical analysis in warfare, 215- 
222 

design and use of individual devices, 
216-218 

general discussion, 215-216 
justification for mathematical sta¬ 
tistics, 218-219 
operational analysis, 218 
recommendations for mathematical 
consultant to air forces, 221-222 
Matrix algebra applied to airborne 
data, 98 

Mechanical lead computing sights, 66- 
69 

class B errors, 68-69 
K-3 sight, 66-68 
K-12 sight, 68 

lateral error computation, 69 
sight equations, 68-69 
“Micro-theory” of warfare, 208 
“Military worth,” general theory of air 
warfare, 200-203 

MPI (mean point of impact) in aerial 
gunnery, 18, 103-104 
Muzzle velocity, effect on dispersion 
pattern, 20 

Muzzle velocity errors in antiaircraft 
fire control systems, 163 

Offensive air warfare, 197-198 
Offset guns in conventional fighters, 
119, 120 

Operational errors in fire control, 94-95 
Operational research groups, recom¬ 
mendations for formation, 207- 
208 

Operational stability, 62-63 
Optical system of gyros, 72-74 
Own-speed plus rate mechanization, 
117-118 

Own-speed sights, 45-56 
approach angle prediction, 45 
class B errors, 55 
definition, 45 
K-ll algebraic sight. 54 




K-13 vector sight, 52-54 
position firing, 50-52 
support fire, 55-56 
tactical considerations, 45 
Own-speed sights, deflection against 
pursuit curves, 46-48 
aerodynamic lead pursuit, 46 
choice of optimum percentage, 47- 
48 

deviated pursuit, 46 
percentage factor variables, 46 
pure pursuit, 46 

Own-speed sights, theory verification, 
48-50 

analytic check, 50 

check of optimum percentages, 49-50 
combat evidence, 48-49 

Pantograph, 2CH computer, 87 
Parallax correction in target bearing, 99 
Parallax unit, 2CH computer, 87 
Peanut (automatic release rocket sight), 
142 

Photography used in airborne assess¬ 
ment programs, 96-97 
Plane of action, 23 

Plaxie (parallax correction mechanism), 
99 

Position firing with own-speed sights, 
50-52 

Position stabilization, 107 
Potentiometer resolver, 2CH computer, 
88 

Predicted position errors, antiaircraft 
fire control system, 162 
Prediction circuits, 164-165 
basic description, 164 
conventional circuit, 164 
response analysis methods, 165 
Sperry A-circuit, 164 
tangential circuit, 164 
Tappert circuit, 164 
Predictor’s time base, antiaircraft fire 
control, 148-149 
Probability 

see also: Damage from antiaircraft 
bursts, probability study 
damage to single aircraft, antiaircraft 
fire, 179-181 

destruction probability, targets, 152 
dry run probability theory, 151 
shot with given dry run will destroy 
target, 153-155 

survival probability estimation pro¬ 
cedure, 155 

Projectile launching, antiaircraft fire, 
168-169 

Projectiles, comparison studies, 187- 
189 

antiaircraft turret installations, 188 
automatic weapons, 188, 189 






INDEX 



249 


B-29 vulnerability from different 
ammunition types, 188, 189 
conditional probability with differ¬ 
ent bombs, 188, 189 
shrapnel shell, 187-189 

Proximity fuzes, distribution of shell 
bursts, 173-176 

Proximity fuzes, effectiveness compared 
with time fuzes, 183-185 

Pursuit curves, 30-48 
angle of attack, 37-39 
bifurcated pursuit, 32-35 
centrifugal force and isogees, 35-36 
combat maneuvers by bomber, 42-43 
deflection against pursuit curves, 46- 
48 

deviation function, 37 
high speed fighter attacking high 
speed bomber, 43-44 
modern warfare applications, 30 
pure and deviated pursuit theory, 31- 
37 

pure pursuit defined, 30 
reasons for investigation, 30-31 
tactical considerations, 42-44 
total load factors, 36-37 
true aerodynamic lead pursuit curve, 
40-42 

Pursuit curves, equation, 31-32, 40-42 
bifurcated pursuit, 32-35 
coordinates, 31 
dynamical equations, 40 
fixed lead pursuit, 32 
kinematical equations, 40 
methods of introducing time as a 
parameter, 35 
pure pursuit, 32 

three dimensional equations, 41-42 
variable lead pursuit, 32 

PUSS (computing sight), 140-141 

Quadratic and linear directors, 158-162 
error prediction, 158-160 
random errors, 161 
theoretical performance, 161-162 
types of directors, 158 

Quantitative analysis, World War II 
aerial warfare, 207-215 
B-29 studies, 209-215 
fighter-plane armaments, 208-209 
“macro” versus “micro” theory of 
warfare, 208 

Quasi-steady errors, fire control, 95 

Radar gunnery aids, 112-115 
airborne gun laying sets, 113 
airborne gun sights (AGS), 113 
airborne range only sets (ARO), 113 
presentation, 112-113 
radar-gyro-sight system, 113-115 
requirements, 112-113 


Radial grid method, tracer stereo¬ 
graphs, 166 

Range determination methods, 96 
Range rates, smoothing possibilities, 59 
Rate deflection of lead computing 
sights, 57-58 
Rate stabilization, 107 
Reference systems, 9-10 
Relative wind, 9 

Remote control, advantages and disad¬ 
vantages, 82 

Research recommendations 

aerial gunnery research unit forma¬ 
tion, 223-225 

fire control development, 107 
mathematical consultant for air 
forces, 221-222 

simulating electronic circuits, 117 
Risk computations, antiaircraft fire 
see Antiaircraft risk computations 
Risks to group of aircraft from flak, 
193-194 


computer requirements, 138-139 
computing sights, 140-142 
hypothetical simple computers, 132- 
133 

lead, 133-139 

operational instability, 138 
pilot’s estimation of variables, 139- 
140 

rate of rotation of sight line, 140 
sighting procedures, 137 
Skid, airborne rockets, 127-128 
Slewing routine, smoothing circuit, 61- 
62 

Slowdown factor, 23 
Small of projectile, motion in, 15-17 
drift, 17 

motion and damping of yaw, 15-16 
windage jump, 16-17 
Smoothing circuit 

see Exponential smoothing circuit 
Space path, methods of determination, 
96 


Rocketry 

see Airborne rockets 
Rotation of coordinates, airborne data, 
97-99 

glook mechanism, 98 
gnomonic computer, 98 
matrix methods, 98 

Shell burst, antiaircraft fire, 169-170 
Shell bursts, distribution, 172-176 
approximations of damage calcula¬ 
tions, 176 

burst surface, 173-176 
circular Gaussian distribution, 18 
effect on damage probability, 182- 
183 

probable density of bursts, 172-173 
proximity fuzes, 173-176 
sensitivity of fuze, 173-176 
spheroidal Gaussian distribution, 173 
time fuzes, 172-173 
Shrapnel shells compared, 187-189 
Siacci coordinates, 9-12 
Sight parameters 

factors effecting choice, 63, 64 
function of range, 115 
physical interpretation, 60-61 
Sight response factor, 62 
Sight systems 

gyroscopic lead computing sights, 69- 
80 

lead-computing sights, 57-81 
mechanical lead computing sights, 
66-69 

own-speed sights, 45-56 
stabilized sight systems, 107-112 
Sighting mechanism, airborne rockets, 
132-142 

basic problem, 132 


Sperry A-circuit, 164 
Sperry K-3 sight, 58 
Sperry S-8B sight system, 110-112 
advantages, 112 

construction and operation, 110-112 
errors caused by aircraft acceler¬ 
ations, 112 

Sperry sights, general discussion, 66 
Stabilization observations made from 
rotating gun platform, 157-158 
Stabilized sight systems, 107-112 
Fairchild S-4 system, 108-110 
Sperry S-8B sight system, 110-112 
stabilization, nature and types, 107 
Stibitz method, tracer stereographs, 166 
Support fire use of own-speed sight, 
55-56 

Survival probability of targets, 152,155 

Tactical-strategic computer (TSC), 
hypothetical military worth 
computer, 201-203 
Tactics, new developments, 119-121 
attack by pacing behind and below, 
119-120 

attack on collision course, 120 
offset gun attacks, 119-121 
upward barrage fire, 121 
Tangential prediction circuit, 164 
Tappert prediction circuit, 164 
Target course curvature, correction 
mechanisms, 117 

Target course curvature, effect on gyro¬ 
scopic sights, 79 

Target destruction probability, 152 
Target speed, conditional probability 
calculations, 177 
TAS (true airspeed), 38 







Three dimensional pursuit curve equa¬ 
tions, 41-42 

Time back method, deflection compu¬ 
tation, 29 
Time fuzes 

distribution of shell bursts, 172-173 
effectiveness compared with prox¬ 
imity fuzes, 183-185 
Time of flight 

calibration concept, 64-65 
fin-stabilized rockets, 127 
fire control computer, 84 
multiplier, 25 
reference systems, 9-10 
relative system, 12-13 
Siacci system, 10-12 
trajectory basic equations, 10-13 
2CH computer, 86, 89 
Torque, gyro systems, 74 
Total load on aircraft, 36-37 
Tracer stereographs 
radial grid method, 166 
Stibit.z method, 166 
Tracking 

aided tracking, 64, 116-117 
amplification factor in tracking 
errors, 62 
equation, 118-119 

errors in angles in antiaircraft fire 
control, 162 
rate formulation, 24 


rate smoothing, 60 
with a fixed sight for airborne 
rockets, 129-130 
Trajectory equations 
basic equations, 10-13 
central station fire control, 83 
Trajectory shift, 37 
Trial angle, ballistic deflection, 13 
Trial fire procedure, 165-166 
Trial gradient in aeroballistics, 20 
True airspeed (TAS), 12, 38-39 
Turret installations, antiaircraft equip¬ 
ment, 188 
Turrets, types, 78 
2CH computer, 82-93 
2CH computer, equations, 83-84 
corrections for angular ballistic and 
parallax, 83-84 
kinematic deflection, 84 
time of flight, 84 

trajectory equation, vector form, 83 
2CH computer, fire control system, 85- 
92 

calibration, 92 
component parts, 87-92 
computer problem, 85 
problem solution method, 85-87 
2CH computer, parts, 87-92 
axis converter, 88 
ballistic unit, 87 
gyroscope units, 89-90 



pantograph, 87 
parallax unit, 87 
potentiometer resolver, 88 
range follow-up motor, 92 
time-of-flight circuit, 86, 89 
total correction motors, 85, 90-92 

Unconditional vulnerability of targets, 
151 

University of Texas testing engine, 107 

Upward barrage fire, 121 

Variance and covariance, vulnerability 
distribution, 154 

Vector method of derivation, deflec¬ 
tion formula, 27-28 

Vector sight, K-13; 52-54 

Vulnerability of targets, conditional 
and unconditional, 151 

Weighted averages, smoothing circuit, 
61 

Windage jump, 16-17 

World War II aerial warfare, quanti¬ 
tative analysis, 207-215 

Yaw, motion and damping, 15-16 

Zero lift line, airborne rockets, 128 


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